Question 1

Solve the differential equation $$10 y ^ { tt } - x y ^ { t } - y = 0$$

Accepted Answer

A)  $y = a _ { 0 } \sum _ { n = 0 } ^ { \infty } \frac { x ^ { 2 n } } { 10 ^ { n } 2 ^ { n } n ! }$, where $a _ { 0 }$ is an arbitrary constant 
B)  $y = a _ { 0 } \sum _ { n = 0 } ^ { \infty } \frac { x ^ { 2 n } } { 10 ^ { n } n ! } + a _ { 1 } \sum _ { n = 0 } ^ { \infty } \frac { x ^ { 2 n + 1 } } { ( n + 1 ) ! }$, where $a _ { 0 }$ and $a _ { 1 }$ are arbitrary constants 
C)  $y = a _ { 0 } \sum _ { n = 0 } ^ { \infty } \frac { x ^ { 2 n } } { 10 ^ { n } 2 ^ { n } n ! } + a _ { 1 } \sum _ { n = 0 } ^ { \infty } \frac { x ^ { 2 n + 1 } } { 10 ^ { n } \cdot 1 \cdot 3 \cdot 5 \cdot 7 \cdots ( 2 n + 1 ) }$, where $a _ { 0 }$ and $a _ { 1 }$ are arbitrary constants 
D)  $y = a _ { 0 } \sum _ { n = 0 } ^ { \infty } \frac { x ^ { 2 n } } { 10 ^ { n } 2 ^ { n + 1 } n ! } + a _ { 1 } \sum _ { n = 0 } ^ { \infty } \frac { x ^ { 2 n + 1 } } { 1 \cdot 3 \cdot 5 \cdot 7 \cdots ( 2 n + 1 ) }$, where $a _ { 0 }$ and $a _ { 1 }$ are arbitrary constants 
E)  $y = a _ { 0 } + a _ { 1 } \sum _ { n = 0 } ^ { \infty } \frac { x ^ { 2 n } } { 10 ^ { n } \cdot 1 \cdot 3 \cdot 5 \cdot 7 \cdots ( 2 n + 1 ) }$, where $a _ { 0 }$ and $a _ { 1 }$ are arbitrary constants 
A)  $y = a _ { 0 } \sum _ { n = 0 } ^ { \infty } \frac { x ^ { 2 n } } { 10 ^ { n } 2 ^ { n } n ! }$, where $a _ { 0 }$ is an arbitrary constant 
B)  $y = a _ { 0 } \sum _ { n = 0 } ^ { \infty } \frac { x ^ { 2 n } } { 10 ^ { n } n ! } + a _ { 1 } \sum _ { n = 0 } ^ { \infty } \frac { x ^ { 2 n + 1 } } { ( n + 1 ) ! }$, where $a _ { 0 }$ and $a _ { 1 }$ are arbitrary constants 
C)  $y = a _ { 0 } \sum _ { n = 0 } ^ { \infty } \frac { x ^ { 2 n } } { 10 ^ { n } 2 ^ { n } n ! } + a _ { 1 } \sum _ { n = 0 } ^ { \infty } \frac { x ^ { 2 n + 1 } } { 10 ^ { n } \cdot 1 \cdot 3 \cdot 5 \cdot 7 \cdots ( 2 n + 1 ) }$, where $a _ { 0 }$ and $a _ { 1 }$ are arbitrary constants 
D)  $y = a _ { 0 } \sum _ { n = 0 } ^ { \infty } \frac { x ^ { 2 n } } { 10 ^ { n } 2 ^ { n + 1 } n ! } + a _ { 1 } \sum _ { n = 0 } ^ { \infty } \frac { x ^ { 2 n + 1 } } { 1 \cdot 3 \cdot 5 \cdot 7 \cdots ( 2 n + 1 ) }$, where $a _ { 0 }$ and $a _ { 1 }$ are arbitrary constants 
E)  $y = a _ { 0 } + a _ { 1 } \sum _ { n = 0 } ^ { \infty } \frac { x ^ { 2 n } } { 10 ^ { n } \cdot 1 \cdot 3 \cdot 5 \cdot 7 \cdots ( 2 n + 1 ) }$, where $a _ { 0 }$ and $a _ { 1 }$ are arbitrary constants C

Question 2

Use the electrical circuit differential equation $\frac { d ^ { 2 } q } { d t ^ { 2 } } + \left( \frac { R } { L } \right) \frac { d q } { d t } + \left( \frac { 1 } { L C } \right) q = \left( \frac { 1 } { L } \right) E ( t )$ where $R = 20$ is the resistance (in ohms), $C = 0.02$ is the capacitance (in farads), $L = 2$ is the inductance (in henrys), $E ( t ) = 14 \sin 6 t$ is the electromotive force (in volts), and $q$ is the charge on the capacitor (in coulombs). Find the charge $q$ as a function of time for the electrical circuit described. Assume that $q ( 0 ) = 0$ and $q ^ { t} ( 0 ) = 0$.

Accepted Answer

A)  $y = \left( \frac { 105 } { 874 } + \frac { 42 } { 71 } t \right) e ^ { - 5 t } - \frac { 105 } { 874 } \cos 6 t - \frac { 77 } { 3,721 } \sin 6 t$ 
B)  $y = \left( \frac { 420 } { 3,721 } + \frac { 42 } { 61 } t \right) e ^ { - 5 t } - \frac { 420 } { 3,721 } \cos 6 t - \frac { 77 } { 3,721 } \sin 6 t$ 
C)  $y = \left( \frac { 105 } { 943 } + \frac { 42 } { 71 } t \right) e ^ { 5 t } - \frac { 105 } { 943 } \cos 6 t - \frac { 77 } { 3,721 } \sin 6 t$ 
D)  $y = \left( \frac { 28 } { 255 } + \frac { 42 } { 61 } t \right) e ^ { - 5 t } - \frac { 28 } { 255 } \cos 6 t - \frac { 77 } { 3,721 } \sin 6 t$ 
E) $y = \left( \frac { 35 } { 306 } + \frac { 42 } { 61 } t \right) e ^ { 5 t } - \frac { 35 } { 306 } \cos 6 t - \frac { 77 } { 3,721 } \sin 6 t$ 
A)  $y = \left( \frac { 105 } { 874 } + \frac { 42 } { 71 } t \right) e ^ { - 5 t } - \frac { 105 } { 874 } \cos 6 t - \frac { 77 } { 3,721 } \sin 6 t$ 
B)  $y = \left( \frac { 420 } { 3,721 } + \frac { 42 } { 61 } t \right) e ^ { - 5 t } - \frac { 420 } { 3,721 } \cos 6 t - \frac { 77 } { 3,721 } \sin 6 t$ 
C)  $y = \left( \frac { 105 } { 943 } + \frac { 42 } { 71 } t \right) e ^ { 5 t } - \frac { 105 } { 943 } \cos 6 t - \frac { 77 } { 3,721 } \sin 6 t$ 
D)  $y = \left( \frac { 28 } { 255 } + \frac { 42 } { 61 } t \right) e ^ { - 5 t } - \frac { 28 } { 255 } \cos 6 t - \frac { 77 } { 3,721 } \sin 6 t$ 
E) $y = \left( \frac { 35 } { 306 } + \frac { 42 } { 61 } t \right) e ^ { 5 t } - \frac { 35 } { 306 } \cos 6 t - \frac { 77 } { 3,721 } \sin 6 t$ B

Question 3

Solve the differential equation $y ^ { tt } - 6 y ^ { t} + 5 y = 20 x \text { by the method of undetermined }$ coefficients.

Accepted Answer

A) $y = C _ { 1 } e ^ { 9 x } + C _ { 2 } e ^ { 5 x } + 4 x - \frac { 24 } { 5 }$ 
B) $y = C _ { 1 } e ^ { 9 x } + C _ { 2 } e ^ { 7 x } + 4 x - \frac { 24 } { 5 }$ 
C) $y = C _ { 1 } e ^ { x } + C _ { 2 } e ^ { 7 x } + 4 x - \frac { 24 } { 5 }$ 
D) $y = C _ { 1 } e ^ { x } + C _ { 2 } e ^ { 5 x } + 4 x + \frac { 24 } { 5 }$ 
E) $y = C _ { 1 } e ^ { 9 x } + C _ { 2 } e ^ { 7 x } + 4 x + \frac { 24 } { 5 }$ 
A) $y = C _ { 1 } e ^ { 9 x } + C _ { 2 } e ^ { 5 x } + 4 x - \frac { 24 } { 5 }$ 
B) $y = C _ { 1 } e ^ { 9 x } + C _ { 2 } e ^ { 7 x } + 4 x - \frac { 24 } { 5 }$ 
C) $y = C _ { 1 } e ^ { x } + C _ { 2 } e ^ { 7 x } + 4 x - \frac { 24 } { 5 }$ 
D) $y = C _ { 1 } e ^ { x } + C _ { 2 } e ^ { 5 x } + 4 x + \frac { 24 } { 5 }$ 
E) $y = C _ { 1 } e ^ { 9 x } + C _ { 2 } e ^ { 7 x } + 4 x + \frac { 24 } { 5 }$ D

Question 4

Use a power series to solve the differential equation $$y ^ { tt } - 25 y = 0$$

Accepted Answer

A)  $y = C _ { 0 } e ^ { 5 x } + C _ { 1 } e ^ { - 5 x }$, where $C _ { 0 }$ and $C _ { 1 }$ are arbitrary constants 
B)  $y = C _ { 0 } x + C _ { 1 } e ^ { 5 x }$, where $C _ { 0 }$ and $C _ { 1 }$ are arbitrary constants 
C)  $y = C _ { 0 } x + C _ { 1 } e ^ { - 5 x }$, where $C _ { 0 }$ and $C _ { 1 }$ are arbitrary constants 
D)  $y = \left( C _ { 0 } + C _ { 1 } x \right) e ^ { - 5 x }$, where $C _ { 0 }$ and $C _ { 1 }$ are arbitrary constants 
E)  $y = \left( C _ { 0 } + C _ { 1 } x \right) e ^ { 5 x }$, where $C _ { 0 }$ and $C _ { 1 }$ are arbitrary constants 
A)  $y = C _ { 0 } e ^ { 5 x } + C _ { 1 } e ^ { - 5 x }$, where $C _ { 0 }$ and $C _ { 1 }$ are arbitrary constants 
B)  $y = C _ { 0 } x + C _ { 1 } e ^ { 5 x }$, where $C _ { 0 }$ and $C _ { 1 }$ are arbitrary constants 
C)  $y = C _ { 0 } x + C _ { 1 } e ^ { - 5 x }$, where $C _ { 0 }$ and $C _ { 1 }$ are arbitrary constants 
D)  $y = \left( C _ { 0 } + C _ { 1 } x \right) e ^ { - 5 x }$, where $C _ { 0 }$ and $C _ { 1 }$ are arbitrary constants 
E)  $y = \left( C _ { 0 } + C _ { 1 } x \right) e ^ { 5 x }$, where $C _ { 0 }$ and $C _ { 1 }$ are arbitrary constants

Question 5

$\text { Use } u ( x , y ) = x ^ { 4 } y \text { as a integrating factor to find the general solution of the }$ differential equation $\left( 5 y ^ { 4 } + 8 x ^ { 3 } y \right) d x + \left( 5 x y ^ { 3 } + 2 x ^ { 4 } \right) d y = 0$

Accepted Answer

A)  $x ^ { 4 } y ^ { 7 } + x ^ { 10 } y ^ { 2 } = C$ 
B)  $x ^ { 5 } y ^ { 5 } + x ^ { 8 } y ^ { 3 } = C$ 
C)  $x ^ { 5 } y ^ { 5 } + x ^ { 8 } y ^ { 2 } = C$ 
D)  $x ^ { 6 } y ^ { 4 } + x ^ { 9 } y ^ { 2 } = C$ 
E)  $x ^ { 6 } y ^ { 4 } + x ^ { 9 } y ^ { 3 } = C$ 
A)  $x ^ { 4 } y ^ { 7 } + x ^ { 10 } y ^ { 2 } = C$ 
B)  $x ^ { 5 } y ^ { 5 } + x ^ { 8 } y ^ { 3 } = C$ 
C)  $x ^ { 5 } y ^ { 5 } + x ^ { 8 } y ^ { 2 } = C$ 
D)  $x ^ { 6 } y ^ { 4 } + x ^ { 9 } y ^ { 2 } = C$ 
E)  $x ^ { 6 } y ^ { 4 } + x ^ { 9 } y ^ { 3 } = C$

Question 6

Find the first six terms of the power series representing independent solutions of the differential equation $\left( x ^ { 2 } + 2 \right) y ^ { tt } + y = 0$.

Accepted Answer

A) $y = a _ { 0 } \left( 1 - \frac { x ^ { 2 } } { 4 } - \frac { x ^ { 4 } } { 32 } - \cdots \right) + a _ { 1 } \left( x - \frac { x ^ { 3 } } { 12 } - \frac { 7 x ^ { 5 } } { 480 } - \cdots \right)$, where $a _ { 0 }$ and $a _ { 1 }$ are arbitrary constants 
B)  $y = a _ { 0 } \left( 1 - \frac { x ^ { 2 } } { 4 } + \frac { x ^ { 4 } } { 96 } - \cdots \right) + a _ { 1 } \left( x - \frac { x ^ { 3 } } { 12 } + \frac { 7 x ^ { 5 } } { 480 } - \cdots \right)$, where $a _ { 0 }$ and $a _ { 1 }$ are arbitrary constants 
C)  $y = a _ { 0 } \left( 1 - \frac { x ^ { 2 } } { 4 } + \frac { x ^ { 4 } } { 32 } - \cdots \right) + a _ { 1 } \left( x - \frac { x ^ { 3 } } { 12 } + \frac { 7 x ^ { 5 } } { 480 } - \cdots \right)$, where $a _ { 0 }$ and $a _ { 1 }$ are arbitrary constants 
D)  $y = a _ { 0 } \left( 1 - \frac { x ^ { 2 } } { 2 } + \frac { x ^ { 4 } } { 8 } - \cdots \right) + a _ { 1 } \left( x - \frac { x ^ { 3 } } { 6 } + \frac { 7 x ^ { 5 } } { 120 } - \cdots \right)$, where $a _ { 0 }$ and $a _ { 1 }$ are arbitrary constants 
E)  $y = a _ { 0 } \left( 1 - \frac { x ^ { 2 } } { 4 } + \frac { x ^ { 4 } } { 16 } - \cdots \right) + a _ { 1 } \left( x - \frac { x ^ { 3 } } { 12 } + \frac { 7 x ^ { 5 } } { 240 } - \cdots \right)$, where $a _ { 0 }$ and $a _ { 1 }$ are arbitrary constants 
A) $y = a _ { 0 } \left( 1 - \frac { x ^ { 2 } } { 4 } - \frac { x ^ { 4 } } { 32 } - \cdots \right) + a _ { 1 } \left( x - \frac { x ^ { 3 } } { 12 } - \frac { 7 x ^ { 5 } } { 480 } - \cdots \right)$, where $a _ { 0 }$ and $a _ { 1 }$ are arbitrary constants 
B)  $y = a _ { 0 } \left( 1 - \frac { x ^ { 2 } } { 4 } + \frac { x ^ { 4 } } { 96 } - \cdots \right) + a _ { 1 } \left( x - \frac { x ^ { 3 } } { 12 } + \frac { 7 x ^ { 5 } } { 480 } - \cdots \right)$, where $a _ { 0 }$ and $a _ { 1 }$ are arbitrary constants 
C)  $y = a _ { 0 } \left( 1 - \frac { x ^ { 2 } } { 4 } + \frac { x ^ { 4 } } { 32 } - \cdots \right) + a _ { 1 } \left( x - \frac { x ^ { 3 } } { 12 } + \frac { 7 x ^ { 5 } } { 480 } - \cdots \right)$, where $a _ { 0 }$ and $a _ { 1 }$ are arbitrary constants 
D)  $y = a _ { 0 } \left( 1 - \frac { x ^ { 2 } } { 2 } + \frac { x ^ { 4 } } { 8 } - \cdots \right) + a _ { 1 } \left( x - \frac { x ^ { 3 } } { 6 } + \frac { 7 x ^ { 5 } } { 120 } - \cdots \right)$, where $a _ { 0 }$ and $a _ { 1 }$ are arbitrary constants 
E)  $y = a _ { 0 } \left( 1 - \frac { x ^ { 2 } } { 4 } + \frac { x ^ { 4 } } { 16 } - \cdots \right) + a _ { 1 } \left( x - \frac { x ^ { 3 } } { 12 } + \frac { 7 x ^ { 5 } } { 240 } - \cdots \right)$, where $a _ { 0 }$ and $a _ { 1 }$ are arbitrary constants

Question 7

Find the interval of convergence for the solution of the differential equation $$y ^ {t} + 3 x y = 0 \text {. }$$

Accepted Answer

A)  $[ 3 , \infty )$ 
B)  $( - 3 , \infty )$ 
C)  $( 0 , \infty )$ 
D)  $( - \infty , \infty )$ 
E)  $[ 0 , \infty )$ 
A)  $[ 3 , \infty )$ 
B)  $( - 3 , \infty )$ 
C)  $( 0 , \infty )$ 
D)  $( - \infty , \infty )$ 
E)  $[ 0 , \infty )$

Question 8

Use Taylor's Theorem to find the first four terms of the series solution of $y ^ { tt } + e ^ { x } y ^ {t } - ( \sin x ) y = 0$ given the initial conditions $y ( 0 ) = - 5$, and $y ^ { t } ( 0 ) = 7$ and use it to calculate $y \left( \frac { 1 } { 4 } \right)$. Round your answer to three decimal places.

Accepted Answer

A)  $y = - 5 + \frac { 7 } { 1 ! } x - \frac { 5 } { 2 ! } x ^ { 2 } - \frac { 7 } { 3 ! } x ^ { 3 } + \cdots , y \left( \frac { 1 } { 4 } \right) \approx - 3.424$ 
B)  $y = - 5 + \frac { 7 } { 1 ! } x - \frac { 5 } { 2 ! } x ^ { 2 } + \frac { 7 } { 3 ! } x ^ { 3 } + \cdots ; y \left( \frac { 1 } { 4 } \right) \approx - 3.388$ 
C)  $y = - 5 + \frac { 7 } { 1 ! } x - \frac { 21 } { 2 ! } x ^ { 2 } - \frac { 10 } { 3 ! } x ^ { 3 } + \cdots , y \left( \frac { 1 } { 4 } \right) \approx - 3.932$ 
D) $y = - 5 + \frac { 7 } { 1 ! } x - \frac { 7 } { 2 ! } x ^ { 2 } + \frac { 5 } { 3 ! } x ^ { 3 } + \cdots , y \left( \frac { 1 } { 4 } \right) \approx - 3.456$ 
E) $y = - 5 + \frac { 7 } { 1 ! } x - \frac { 7 } { 2 ! } x ^ { 2 } - \frac { 5 } { 3 ! } x ^ { 3 } + \cdots ; y \left( \frac { 1 } { 4 } \right) \approx - 3.482$ 
A)  $y = - 5 + \frac { 7 } { 1 ! } x - \frac { 5 } { 2 ! } x ^ { 2 } - \frac { 7 } { 3 ! } x ^ { 3 } + \cdots , y \left( \frac { 1 } { 4 } \right) \approx - 3.424$ 
B)  $y = - 5 + \frac { 7 } { 1 ! } x - \frac { 5 } { 2 ! } x ^ { 2 } + \frac { 7 } { 3 ! } x ^ { 3 } + \cdots ; y \left( \frac { 1 } { 4 } \right) \approx - 3.388$ 
C)  $y = - 5 + \frac { 7 } { 1 ! } x - \frac { 21 } { 2 ! } x ^ { 2 } - \frac { 10 } { 3 ! } x ^ { 3 } + \cdots , y \left( \frac { 1 } { 4 } \right) \approx - 3.932$ 
D) $y = - 5 + \frac { 7 } { 1 ! } x - \frac { 7 } { 2 ! } x ^ { 2 } + \frac { 5 } { 3 ! } x ^ { 3 } + \cdots , y \left( \frac { 1 } { 4 } \right) \approx - 3.456$ 
E) $y = - 5 + \frac { 7 } { 1 ! } x - \frac { 7 } { 2 ! } x ^ { 2 } - \frac { 5 } { 3 ! } x ^ { 3 } + \cdots ; y \left( \frac { 1 } { 4 } \right) \approx - 3.482$

Question 9

Find the interval of convergence for the solution of the differential equation $$3 y ^ { tt } - x y ^ { t } - y = 0$$

Accepted Answer

A)  $[ 3 , \infty )$ 
B)  $[ 0 , \infty )$ 
C)  $( 0 , \infty )$ 
D)  $( - 3 , \infty )$ 
E)  $( - \infty , \infty )$ 
A)  $[ 3 , \infty )$ 
B)  $[ 0 , \infty )$ 
C)  $( 0 , \infty )$ 
D)  $( - 3 , \infty )$ 
E)  $( - \infty , \infty )$

Question 10

Solve the differential equation $$y ^ { tt } - 9 x y ^ { t } = 0$$

Accepted Answer

A)  $y = a _ { 0 } + a _ { 1 } \sum _ { n = 0 } ^ { \infty } \frac { 9 ^ { n } x ^ { 2 n + 1 } } { 2 ^ { n } n ! }$, where $a _ { 0 }$ and $a _ { 1 }$ are arbitrary constants 
B)  $y = a _ { 0 } + a _ { 1 } \sum _ { n = 0 } ^ { \infty } \frac { 9 ^ { n + 1 } x ^ { 2 n } } { 2 ^ { n } ( 2 n + 1 ) ! }$, where $a _ { 0 }$ and $a _ { 1 }$ are arbitrary constants 
C)  $y = a _ { 0 } + a _ { 1 } \sum _ { n = 0 } ^ { \infty } \frac { 9 ^ { n } x ^ { 2 n + 1 } } { 2 ^ { n } n ! ( 2 n + 1 ) }$, where $a _ { 0 }$ and $a _ { 1 }$ are arbitrary constants 
D)  $y = a _ { 0 } + a _ { 1 } \sum _ { n = 0 } ^ { \infty } \frac { 9 ^ { n + 1 } x ^ { 2 n } } { 2 ^ { n } n ! }$, where $a _ { 0 }$ and $a _ { 1 }$ are arbitrary constants 
E)  $y = a _ { 0 } + a _ { 1 } \sum _ { n = 0 } ^ { \infty } \frac { 9 ^ { n + 1 } x ^ { 2 n + 1 } } { 2 ^ { n } n ! ( 2 n + 1 ) }$, where $a _ { 0 }$ and $a _ { 1 }$ are arbitrary constants 
A)  $y = a _ { 0 } + a _ { 1 } \sum _ { n = 0 } ^ { \infty } \frac { 9 ^ { n } x ^ { 2 n + 1 } } { 2 ^ { n } n ! }$, where $a _ { 0 }$ and $a _ { 1 }$ are arbitrary constants 
B)  $y = a _ { 0 } + a _ { 1 } \sum _ { n = 0 } ^ { \infty } \frac { 9 ^ { n + 1 } x ^ { 2 n } } { 2 ^ { n } ( 2 n + 1 ) ! }$, where $a _ { 0 }$ and $a _ { 1 }$ are arbitrary constants 
C)  $y = a _ { 0 } + a _ { 1 } \sum _ { n = 0 } ^ { \infty } \frac { 9 ^ { n } x ^ { 2 n + 1 } } { 2 ^ { n } n ! ( 2 n + 1 ) }$, where $a _ { 0 }$ and $a _ { 1 }$ are arbitrary constants 
D)  $y = a _ { 0 } + a _ { 1 } \sum _ { n = 0 } ^ { \infty } \frac { 9 ^ { n + 1 } x ^ { 2 n } } { 2 ^ { n } n ! }$, where $a _ { 0 }$ and $a _ { 1 }$ are arbitrary constants 
E)  $y = a _ { 0 } + a _ { 1 } \sum _ { n = 0 } ^ { \infty } \frac { 9 ^ { n + 1 } x ^ { 2 n + 1 } } { 2 ^ { n } n ! ( 2 n + 1 ) }$, where $a _ { 0 }$ and $a _ { 1 }$ are arbitrary constants

Question 11

Find the series solution of the differential equation $y ^ { tt } - 7 x y ^ { t } = 0 \text { given the initial }$ conditions $y ( 0 ) = 0$ and $y ^ { t } ( 0 ) = 4$

Accepted Answer

A)  $y = 4 \left( 1 + \sum _ { n = 0 } ^ { \infty } \frac { 7 ^ { n + 1 } x ^ { 2 n + 1 } } { 2 ^ { n } n ! ( 2 n + 1 ) } \right)$ 
B) $y = 4 \sum _ { n = 0 } ^ { \infty } \frac { 7 ^ { n } x ^ { 2 n + 1 } } { 2 ^ { n } n ! ( 2 n + 1 ) }$ 
C) $y = 4 \sum _ { n = 0 } ^ { \infty } \frac { 7 ^ { n + 1 } x ^ { 2 n + 1 } } { 2 ^ { n } ( 2 n + 1 ) ! }$ 
D) $y = 4 + \sum _ { n = 0 } ^ { \infty } \frac { 7 ^ { n } x ^ { 2 n + 1 } } { 2 ^ { n } ( 2 n + 1 ) ! }$ 
E) $y = 4 + \sum _ { n = 0 } ^ { \infty } \frac { 7 ^ { n } x ^ { 2 n + 1 } } { 2 ^ { n } n ! ( 2 n + 1 ) }$ 
A)  $y = 4 \left( 1 + \sum _ { n = 0 } ^ { \infty } \frac { 7 ^ { n + 1 } x ^ { 2 n + 1 } } { 2 ^ { n } n ! ( 2 n + 1 ) } \right)$ 
B) $y = 4 \sum _ { n = 0 } ^ { \infty } \frac { 7 ^ { n } x ^ { 2 n + 1 } } { 2 ^ { n } n ! ( 2 n + 1 ) }$ 
C) $y = 4 \sum _ { n = 0 } ^ { \infty } \frac { 7 ^ { n + 1 } x ^ { 2 n + 1 } } { 2 ^ { n } ( 2 n + 1 ) ! }$ 
D) $y = 4 + \sum _ { n = 0 } ^ { \infty } \frac { 7 ^ { n } x ^ { 2 n + 1 } } { 2 ^ { n } ( 2 n + 1 ) ! }$ 
E) $y = 4 + \sum _ { n = 0 } ^ { \infty } \frac { 7 ^ { n } x ^ { 2 n + 1 } } { 2 ^ { n } n ! ( 2 n + 1 ) }$

Question 12

Solve the differential equation $144 y ^ {tt} - 24 y ^ { t } + y = 12 \left( x + e ^ { x } \right) \text { by the method of }$ undetermined coefficients.

Accepted Answer

A)  $y = \left( C _ { 1 } + C _ { 2 } x \right) e ^ { 1 / 2 x } + 288 + 12 x + \frac { 12 } { 121 } e ^ { x }$ 
B)  $y = \left( C _ { 1 } + C _ { 2 } x \right) e ^ { 1 / 12 x } + 288 + 13 x + \frac { 12 } { 121 } e ^ { x }$ 
C) $y = \left( C _ { 1 } + C _ { 2 } x \right) e ^ { 1 / 12 x } + 288 + 5 x + \frac { 12 } { 121 } e ^ { x }$ 
D) $y = \left( C _ { 1 } + C _ { 2 } x \right) e ^ { 1 / 12 x } + 288 + 12 x + \frac { 12 } { 121 } e ^ { x }$ 
E) $y = \left( C _ { 1 } + C _ { 2 } x \right) e ^ { 1 / 2 x } + 288 + 2 x + \frac { 12 } { 121 } e ^ { x }$ 
A)  $y = \left( C _ { 1 } + C _ { 2 } x \right) e ^ { 1 / 2 x } + 288 + 12 x + \frac { 12 } { 121 } e ^ { x }$ 
B)  $y = \left( C _ { 1 } + C _ { 2 } x \right) e ^ { 1 / 12 x } + 288 + 13 x + \frac { 12 } { 121 } e ^ { x }$ 
C) $y = \left( C _ { 1 } + C _ { 2 } x \right) e ^ { 1 / 12 x } + 288 + 5 x + \frac { 12 } { 121 } e ^ { x }$ 
D) $y = \left( C _ { 1 } + C _ { 2 } x \right) e ^ { 1 / 12 x } + 288 + 12 x + \frac { 12 } { 121 } e ^ { x }$ 
E) $y = \left( C _ { 1 } + C _ { 2 } x \right) e ^ { 1 / 2 x } + 288 + 2 x + \frac { 12 } { 121 } e ^ { x }$

Question 13

Use Taylor's Theorem to find the first eight terms of the series solution of $y ^ { tt } - 2 x y ^ { t } + y = 0$ given the initial conditions $y ( 0 ) = 1 , y ^ { t} ( 0 ) = 4$ and use it to calculate $y \left( \frac { 1 } { 3 } \right)$.
Round your answer to three decimal places.

Accepted Answer

A)  $y = 1 + \frac { 4 } { 1 ! } x - \frac { 1 } { 2 ! } x ^ { 2 } + \frac { 4 } { 3 ! } x ^ { 3 } + \frac { 3 } { 4 ! } x ^ { 4 } + \frac { 20 } { 5 ! } x ^ { 5 } + \frac { 21 } { 6 ! } x ^ { 6 } + \frac { 180 } { 7 ! } x ^ { 7 } + \cdots ; y \left( \frac { 1 } { 3 } \right) \leqslant 2.305$ 
B)  $y = 1 + \frac { 4 } { 1 ! } x - \frac { 1 } { 2 ! } x ^ { 2 } + \frac { 4 } { 3 ! } x ^ { 3 } - \frac { 3 } { 4 ! } x ^ { 4 } + \frac { 20 } { 5 ! } x ^ { 5 } - \frac { 21 } { 6 ! } x ^ { 6 } + \frac { 180 } { 7 ! } x ^ { 7 } + \cdots ; y \left( \frac { 1 } { 3 } \right) \approx 2.302$ 
C) $y = 1 - \frac { 4 } { 1 ! } x + \frac { 2 } { 2 ! } x ^ { 2 } - \frac { 4 } { 3 ! } x ^ { 3 } + \frac { 3 } { 4 ! } x ^ { 4 } - \frac { 4 } { 5 ! } x ^ { 5 } + \frac { 4 } { 6 ! } x ^ { 6 } - \frac { 4 } { 7 ! } x ^ { 7 } + \cdots ; y \left( \frac { 1 } { 3 } \right) \approx - 0.468$ 
D) 
$y = 1 + \frac { 4 } { 1 ! } x + \frac { 1 } { 2 ! } x ^ { 2 } + \frac { 12 } { 3 ! } x ^ { 3 } + \frac { 3 } { 4 ! } x ^ { 4 } + \frac { 28 } { 5 ! } x ^ { 5 } - \frac { 84 } { 6 ! } x ^ { 6 } + \frac { 45 } { 7 ! } x ^ { 7 } + \cdots , y \left( \frac { 1 } { 3 } \right) \approx 2.465$ 
E) 
$y = 1 + \frac { 4 } { 1 ! } x - \frac { 1 } { 2 ! } x ^ { 2 } + \frac { 12 } { 3 ! } x ^ { 3 } - \frac { 3 } { 4 ! } x ^ { 4 } + \frac { 28 } { 5 ! } x ^ { 5 } - \frac { 84 } { 6 ! } x ^ { 6 } + \frac { 45 } { 7 ! } x ^ { 7 } + \cdots , y \left( \frac { 1 } { 3 } \right) \approx 2.351$ 
A)  $y = 1 + \frac { 4 } { 1 ! } x - \frac { 1 } { 2 ! } x ^ { 2 } + \frac { 4 } { 3 ! } x ^ { 3 } + \frac { 3 } { 4 ! } x ^ { 4 } + \frac { 20 } { 5 ! } x ^ { 5 } + \frac { 21 } { 6 ! } x ^ { 6 } + \frac { 180 } { 7 ! } x ^ { 7 } + \cdots ; y \left( \frac { 1 } { 3 } \right) \leqslant 2.305$ 
B)  $y = 1 + \frac { 4 } { 1 ! } x - \frac { 1 } { 2 ! } x ^ { 2 } + \frac { 4 } { 3 ! } x ^ { 3 } - \frac { 3 } { 4 ! } x ^ { 4 } + \frac { 20 } { 5 ! } x ^ { 5 } - \frac { 21 } { 6 ! } x ^ { 6 } + \frac { 180 } { 7 ! } x ^ { 7 } + \cdots ; y \left( \frac { 1 } { 3 } \right) \approx 2.302$ 
C) $y = 1 - \frac { 4 } { 1 ! } x + \frac { 2 } { 2 ! } x ^ { 2 } - \frac { 4 } { 3 ! } x ^ { 3 } + \frac { 3 } { 4 ! } x ^ { 4 } - \frac { 4 } { 5 ! } x ^ { 5 } + \frac { 4 } { 6 ! } x ^ { 6 } - \frac { 4 } { 7 ! } x ^ { 7 } + \cdots ; y \left( \frac { 1 } { 3 } \right) \approx - 0.468$ 
D) 
$y = 1 + \frac { 4 } { 1 ! } x + \frac { 1 } { 2 ! } x ^ { 2 } + \frac { 12 } { 3 ! } x ^ { 3 } + \frac { 3 } { 4 ! } x ^ { 4 } + \frac { 28 } { 5 ! } x ^ { 5 } - \frac { 84 } { 6 ! } x ^ { 6 } + \frac { 45 } { 7 ! } x ^ { 7 } + \cdots , y \left( \frac { 1 } { 3 } \right) \approx 2.465$ 
E) 
$y = 1 + \frac { 4 } { 1 ! } x - \frac { 1 } { 2 ! } x ^ { 2 } + \frac { 12 } { 3 ! } x ^ { 3 } - \frac { 3 } { 4 ! } x ^ { 4 } + \frac { 28 } { 5 ! } x ^ { 5 } - \frac { 84 } { 6 ! } x ^ { 6 } + \frac { 45 } { 7 ! } x ^ { 7 } + \cdots , y \left( \frac { 1 } { 3 } \right) \approx 2.351$

Question 14

Find the particular solution of the differential equation $\frac { w } { g } y ^ { tt } ( t ) + b y ^ { t } ( t ) + k y ( t ) = \frac { w } { g } F ( t )$ for the oscillating motion of an object on the end of a spring. In the equation, $y$ is the displacement from equilibrium (positive direction is downward) measured in feet, and $t$ is the time in seconds (see figure). The constant $w = 32$ is the weight of the object, $g = 72$ is the acceleration due to gravity, $b = 0$ is the magnitude of the resistance to the motion, $k = 64$ is the spring constant from Hooke's Law, $F ( t ) = 64 \sin 6 t$ is the acceleration imposed on the system, $y ( 0 ) = \frac { 1 } { 6 }$ and $y ^ { t } ( 0 ) = 0$

Accepted Answer

A)  $y = \frac { 1 } { 6 } \cos 12 t - \frac { 8 } { 27 } \sin 12 t + \frac { 14 } { 27 } \sin 6 t$ 
B)  $y = \frac { 1 } { 12 } \cos 12 t - \frac { 16 } { 63 } \sin 12 t + \frac { 14 } { 27 } \cos 6 t$ 
C)  $y = \frac { 1 } { 12 } \cos 12 t - \frac { 16 } { 63 } \sin 12 t - \frac { 14 } { 27 } \sin 6 t$ 
D)  $y = \frac { 1 } { 6 } \cos 12 t - \frac { 8 } { 27 } \sin 12 t + \frac { 16 } { 27 } \cos 6 t$ 
E)  $y = \frac { 1 } { 6 } \cos 12 t - \frac { 8 } { 27 } \sin 12 t + \frac { 16 } { 27 } \sin 6 t$ 
A)  $y = \frac { 1 } { 6 } \cos 12 t - \frac { 8 } { 27 } \sin 12 t + \frac { 14 } { 27 } \sin 6 t$ 
B)  $y = \frac { 1 } { 12 } \cos 12 t - \frac { 16 } { 63 } \sin 12 t + \frac { 14 } { 27 } \cos 6 t$ 
C)  $y = \frac { 1 } { 12 } \cos 12 t - \frac { 16 } { 63 } \sin 12 t - \frac { 14 } { 27 } \sin 6 t$ 
D)  $y = \frac { 1 } { 6 } \cos 12 t - \frac { 8 } { 27 } \sin 12 t + \frac { 16 } { 27 } \cos 6 t$ 
E)  $y = \frac { 1 } { 6 } \cos 12 t - \frac { 8 } { 27 } \sin 12 t + \frac { 16 } { 27 } \sin 6 t$

Question 15

$\text { Find the integrating factor of the differential equation } ( x + 3 y ) d x - 3 \cot x d y = 0 \text { that }$ is a function of x or y alone.

Accepted Answer

A)  $1 + 3 \cot x$ 
B)  $\cos x$ 
C)  $\sin x$ 
D)  $- 3 \cos x$ 
E)  $- 3 \sin x$ 
A)  $1 + 3 \cot x$ 
B)  $\cos x$ 
C)  $\sin x$ 
D)  $- 3 \cos x$ 
E)  $- 3 \sin x$

Question 16

Find the first eight terms of the power series representing independent solutions of the differential equation $y ^ { tt } + 11 x ^ { 2 } y = 0$.

Accepted Answer

A)  $y = a _ { 0 } \left( 1 - \frac { x ^ { 4 } } { 4 ! } + \frac { x ^ { 8 } } { 8 ! } - \frac { x ^ { 12 } } { 12 ! } + \cdots \right) + a _ { 1 } \left( \frac { x } { 1 ! } - \frac { x ^ { 5 } } { 5 ! } + \frac { x ^ { 9 } } { 9 ! } - \frac { x ^ { 13 } } { 13 ! } + \cdots \right)$, where $a _ { 0 }$ and $a _ { 1 }$ are arbitrary constants 
B) $\begin{aligned}
y = & a _ { 0 } \left( 1 - \frac { 11 \cdot x ^ { 4 } } { 4 \cdot 3 } + \frac { 11 ^ { 2 } \cdot x ^ { 8 } } { 8 \cdot 7 \cdot 4 \cdot 3 } - \frac { 11 ^ { 3 } \cdot x ^ { 12 } } { 12 \cdot 11 \cdot 8 \cdot 7 \cdot 4 \cdot 3 } + \cdots \right) \
& + a _ { 1 } \left( x - \frac { 11 \cdot x ^ { 5 } } { 5 \cdot 4 } + \frac { 11 ^ { 2 } \cdot x ^ { 7 } } { 9 \cdot 8 \cdot 5 \cdot 4 } - \frac { 11 ^ { 3 } \cdot x ^ { 9 } } { 13 \cdot 12 \cdot 9 \cdot 8 \cdot 5 \cdot 4 } + \cdots \right)
\end{aligned}$
where $a _ { 0 }$ and $a _ { 1 }$ are arbitrary constants 
C)  $y = a _ { 0 } \left( 1 - \frac { 11 \cdot x ^ { 4 } } { 2 ! } + \frac { 11 ^ { 2 } \cdot x ^ { 8 } } { 6 ! } - \frac { 11 ^ { 3 } \cdot x ^ { 12 } } { 10 ! } + \cdots \right) + a _ { 1 } \left( \frac { x } { 1 ! } - \frac { 11 \cdot x ^ { 5 } } { 5 ! } + \frac { 11 ^ { 2 } \cdot x ^ { 9 } } { 7 ! } - \frac { 11 ^ { 3 } \cdot x ^ { 13 } } { 11 ! } + \cdots \right)$, where $a _ { 0 }$ and $a _ { 1 }$ are arbitrary constants 
D)  $y = a _ { 0 } \left( 1 - \frac { x ^ { 2 } } { 2 ! } - \frac { x ^ { 6 } } { 6 ! } - \frac { x ^ { 10 } } { 10 ! } + \cdots \right) + a _ { 1 } \left( \frac { x } { 1 ! } - \frac { x ^ { 3 } } { 3 ! } - \frac { x ^ { 7 } } { 7 ! } - \frac { x ^ { 11 } } { 11 ! } + \cdots \right)$, where $a _ { 0 }$ and $a _ { 1 }$ are arbitrary constants 
E) 
$y = a _ { 0 } \left( 1 - \frac { 11 \cdot x ^ { 4 } } { 4 \cdot 3 } - \frac { 11 ^ { 2 } \cdot x ^ { 8 } } { 8 \cdot 7 } - \frac { 11 ^ { 3 } \cdot x ^ { 12 } } { 12 \cdot 11 } + \cdots \right) + a _ { 1 } \left( x - \frac { 11 \cdot x ^ { 5 } } { 5 \cdot 4 } - \frac { 11 ^ { 2 } \cdot x ^ { 9 } } { 9 \cdot 8 } - \frac { 11 ^ { 3 } \cdot x ^ { 13 } } { 13 \cdot 12 } + \cdots \right)$ where $a _ { 0 }$ and $a _ { 1 }$ are arbitrary constants 
A)  $y = a _ { 0 } \left( 1 - \frac { x ^ { 4 } } { 4 ! } + \frac { x ^ { 8 } } { 8 ! } - \frac { x ^ { 12 } } { 12 ! } + \cdots \right) + a _ { 1 } \left( \frac { x } { 1 ! } - \frac { x ^ { 5 } } { 5 ! } + \frac { x ^ { 9 } } { 9 ! } - \frac { x ^ { 13 } } { 13 ! } + \cdots \right)$, where $a _ { 0 }$ and $a _ { 1 }$ are arbitrary constants 
B) $\begin{aligned}
y = & a _ { 0 } \left( 1 - \frac { 11 \cdot x ^ { 4 } } { 4 \cdot 3 } + \frac { 11 ^ { 2 } \cdot x ^ { 8 } } { 8 \cdot 7 \cdot 4 \cdot 3 } - \frac { 11 ^ { 3 } \cdot x ^ { 12 } } { 12 \cdot 11 \cdot 8 \cdot 7 \cdot 4 \cdot 3 } + \cdots \right) \
& + a _ { 1 } \left( x - \frac { 11 \cdot x ^ { 5 } } { 5 \cdot 4 } + \frac { 11 ^ { 2 } \cdot x ^ { 7 } } { 9 \cdot 8 \cdot 5 \cdot 4 } - \frac { 11 ^ { 3 } \cdot x ^ { 9 } } { 13 \cdot 12 \cdot 9 \cdot 8 \cdot 5 \cdot 4 } + \cdots \right)
\end{aligned}$
where $a _ { 0 }$ and $a _ { 1 }$ are arbitrary constants 
C)  $y = a _ { 0 } \left( 1 - \frac { 11 \cdot x ^ { 4 } } { 2 ! } + \frac { 11 ^ { 2 } \cdot x ^ { 8 } } { 6 ! } - \frac { 11 ^ { 3 } \cdot x ^ { 12 } } { 10 ! } + \cdots \right) + a _ { 1 } \left( \frac { x } { 1 ! } - \frac { 11 \cdot x ^ { 5 } } { 5 ! } + \frac { 11 ^ { 2 } \cdot x ^ { 9 } } { 7 ! } - \frac { 11 ^ { 3 } \cdot x ^ { 13 } } { 11 ! } + \cdots \right)$, where $a _ { 0 }$ and $a _ { 1 }$ are arbitrary constants 
D)  $y = a _ { 0 } \left( 1 - \frac { x ^ { 2 } } { 2 ! } - \frac { x ^ { 6 } } { 6 ! } - \frac { x ^ { 10 } } { 10 ! } + \cdots \right) + a _ { 1 } \left( \frac { x } { 1 ! } - \frac { x ^ { 3 } } { 3 ! } - \frac { x ^ { 7 } } { 7 ! } - \frac { x ^ { 11 } } { 11 ! } + \cdots \right)$, where $a _ { 0 }$ and $a _ { 1 }$ are arbitrary constants 
E) 
$y = a _ { 0 } \left( 1 - \frac { 11 \cdot x ^ { 4 } } { 4 \cdot 3 } - \frac { 11 ^ { 2 } \cdot x ^ { 8 } } { 8 \cdot 7 } - \frac { 11 ^ { 3 } \cdot x ^ { 12 } } { 12 \cdot 11 } + \cdots \right) + a _ { 1 } \left( x - \frac { 11 \cdot x ^ { 5 } } { 5 \cdot 4 } - \frac { 11 ^ { 2 } \cdot x ^ { 9 } } { 9 \cdot 8 } - \frac { 11 ^ { 3 } \cdot x ^ { 13 } } { 13 \cdot 12 } + \cdots \right)$ where $a _ { 0 }$ and $a _ { 1 }$ are arbitrary constants

Question 17

Find the integrating factor of the differential equation $\left( 3 x ^ { 3 } + 2 y \right) d x - x d y = 0 \text { that is }$ a function of x or y alone.

Accepted Answer

A)  $\frac { 1 } { x ^ { - 3 } }$ 
B)  $- \frac { 3 } { x }$ 
C)  $- \frac { 2 } { x }$ 
D)  $\frac { 1 } { x ^ { - 4 } }$ 
E)  $\frac { 1 } { x ^ { 3 } }$ 
A)  $\frac { 1 } { x ^ { - 3 } }$ 
B)  $- \frac { 3 } { x }$ 
C)  $- \frac { 2 } { x }$ 
D)  $\frac { 1 } { x ^ { - 4 } }$ 
E)  $\frac { 1 } { x ^ { 3 } }$

Question 18

Find the particular solution of the differential equation $\frac { 3 y } { x - 8 } d x + [ 3 \ln ( x - 8 ) + 10 y ] d y = 0$ that satisfies the initial condition $y ( 9 ) = 2$

Accepted Answer

A)  $3 y \ln ( x - 8 ) + 5 y = 10$ 
B)  $3 y \ln ( x - 8 ) + 5 y ^ { 2 } = 20$ 
C)  $3 y ( x - 8 ) + 10 y = 26$ 
D)  $3 y \ln ( x - 8 ) + 10 y = 20$ 
E)  $\frac { 3 y } { ( x - 8 ) } + 5 y ^ { 2 } = 26$ 
A)  $3 y \ln ( x - 8 ) + 5 y = 10$ 
B)  $3 y \ln ( x - 8 ) + 5 y ^ { 2 } = 20$ 
C)  $3 y ( x - 8 ) + 10 y = 26$ 
D)  $3 y \ln ( x - 8 ) + 10 y = 20$ 
E)  $\frac { 3 y } { ( x - 8 ) } + 5 y ^ { 2 } = 26$

Question 19

Find the particular solution of the differential equation 
$e ^ { 8 x } ( \sin 8 y d x + \cos 8 y d y ) = 0$ that satisfies the initial condition $y ( 0 ) = \pi$.

Accepted Answer

A)  $e ^ { 8 x } \sin 8 y = 1$ 
B)  $e ^ { 8 x } \sin 8 y = e ^ { 8 x }$ 
C)  $e ^ { 8 x } \cos 8 y = 0$ 
D)  $e ^ { 8 x } \cos 8 y = 1$ 
E)  $e ^ { 8 x } \sin 8 y = 0$ 
A)  $e ^ { 8 x } \sin 8 y = 1$ 
B)  $e ^ { 8 x } \sin 8 y = e ^ { 8 x }$ 
C)  $e ^ { 8 x } \cos 8 y = 0$ 
D)  $e ^ { 8 x } \cos 8 y = 1$ 
E)  $e ^ { 8 x } \sin 8 y = 0$

Question 20

Solve the differential equation $x ^ { 2 } y ^ { tt } - x y ^ { t } + y = 23 x \ln x \text { given that } y _ { 1 } = 9 x \text { and }$ $y _ { 2 } = 9 x \ln x$ are solutions of the corresponding homogeneous equation.

Accepted Answer

A) 
$y = C _ { 1 } 9 x + C _ { 2 } 9 x \ln x - \frac { 23 } { 45 } x ( \ln x ) ^ { 4 }$ 
B) 
$y = C _ { 1 } 9 x + C _ { 2 } 9 x \ln x - \frac { 23 } { 63 } x ( \ln x ) ^ { 4 }$ 
C) 
$y = C _ { 1 } 9 x + C _ { 2 } 9 x \ln x + \frac { 23 } { 90 } x ( \ln x ) ^ { 2 }$ 
D) 
$y = C _ { 1 } 9 x + C _ { 2 } 9 x \ln x + \frac { 23 } { 72 } x ( \ln x ) ^ { 2 }$ 
E) 
$y = C _ { 1 } x + C _ { 2 } x \ln x + \frac { 23 } { 6 } x ( \ln x ) ^ { 3 }$ 
A) 
$y = C _ { 1 } 9 x + C _ { 2 } 9 x \ln x - \frac { 23 } { 45 } x ( \ln x ) ^ { 4 }$ 
B) 
$y = C _ { 1 } 9 x + C _ { 2 } 9 x \ln x - \frac { 23 } { 63 } x ( \ln x ) ^ { 4 }$ 
C) 
$y = C _ { 1 } 9 x + C _ { 2 } 9 x \ln x + \frac { 23 } { 90 } x ( \ln x ) ^ { 2 }$ 
D) 
$y = C _ { 1 } 9 x + C _ { 2 } 9 x \ln x + \frac { 23 } { 72 } x ( \ln x ) ^ { 2 }$ 
E) 
$y = C _ { 1 } x + C _ { 2 } x \ln x + \frac { 23 } { 6 } x ( \ln x ) ^ { 3 }$

Solve the differential equation $10 y ^ { tt } - x y ^ { t } - y = 0$

Solve the differential equation $y ^ { tt } - 6 y ^ { t} + 5 y = 20 x \text { by the method of undetermined }$ coefficients.

Use a power series to solve the differential equation $y ^ { tt } - 25 y = 0$

$\text { Use } u ( x , y ) = x ^ { 4 } y \text { as a integrating factor to find the general solution of the }$ differential equation $\left( 5 y ^ { 4 } + 8 x ^ { 3 } y \right) d x + \left( 5 x y ^ { 3 } + 2 x ^ { 4 } \right) d y = 0$

Find the first six terms of the power series representing independent solutions of the differential equation $\left( x ^ { 2 } + 2 \right) y ^ { tt } + y = 0$ .

Find the interval of convergence for the solution of the differential equation $y ^ {t} + 3 x y = 0 \text {. }$

Find the interval of convergence for the solution of the differential equation $3 y ^ { tt } - x y ^ { t } - y = 0$

Solve the differential equation $y ^ { tt } - 9 x y ^ { t } = 0$

Find the series solution of the differential equation $y ^ { tt } - 7 x y ^ { t } = 0 \text { given the initial }$ conditions $y ( 0 ) = 0$ and $y ^ { t } ( 0 ) = 4$

Solve the differential equation $144 y ^ {tt} - 24 y ^ { t } + y = 12 \left( x + e ^ { x } \right) \text { by the method of }$ undetermined coefficients.

Use Taylor's Theorem to find the first eight terms of the series solution of $y ^ { tt } - 2 x y ^ { t } + y = 0$ given the initial conditions $y ( 0 ) = 1 , y ^ { t} ( 0 ) = 4$ and use it to calculate $y \left( \frac { 1 } { 3 } \right)$ . Round your answer to three decimal places.

$\text { Find the integrating factor of the differential equation } ( x + 3 y ) d x - 3 \cot x d y = 0 \text { that }$ is a function of x or y alone.

Find the first eight terms of the power series representing independent solutions of the differential equation $y ^ { tt } + 11 x ^ { 2 } y = 0$ .

Find the integrating factor of the differential equation $\left( 3 x ^ { 3 } + 2 y \right) d x - x d y = 0 \text { that is }$ a function of x or y alone.

Find the particular solution of the differential equation $\frac { 3 y } { x - 8 } d x + [ 3 \ln ( x - 8 ) + 10 y ] d y = 0$ that satisfies the initial condition $y ( 9 ) = 2$

Find the particular solution of the differential equation $e ^ { 8 x } ( \sin 8 y d x + \cos 8 y d y ) = 0$ that satisfies the initial condition $y ( 0 ) = \pi$ .

Solve the differential equation $x ^ { 2 } y ^ { tt } - x y ^ { t } + y = 23 x \ln x \text { given that } y _ { 1 } = 9 x \text { and }$ $y _ { 2 } = 9 x \ln x$ are solutions of the corresponding homogeneous equation.

Graphs and Models

A Preview of Calculus

The Derivative and the Tangent Line Problem

Extrema on an Interval

Antiderivatives and Indefinite Integration

Slope Fields and Eulers Method

Area of a Region Between Two Curves

Basic Integration Rules

Sequences

Conics and Calculus

Vectors in the Plane

Vector-Valued Functions

Introduction to Functions of Several Variables

Iterated Integrals and Area in the Plane

Vector Fields

Filters

Exam 16: Exact First-Order Equations

Solve the differential equation 10ytt−xyt−y=010 y ^ { tt } - x y ^ { t } - y = 010ytt−xyt−y=0

Solve the differential equation ytt−6yt+5y=20x by the method of undetermined y ^ { tt } - 6 y ^ { t} + 5 y = 20 x \text { by the method of undetermined }ytt−6yt+5y=20x by the method of undetermined coefficients.

Use a power series to solve the differential equation ytt−25y=0y ^ { tt } - 25 y = 0ytt−25y=0

Find the first six terms of the power series representing independent solutions of the differential equation (x2+2)ytt+y=0\left( x ^ { 2 } + 2 \right) y ^ { tt } + y = 0(x2+2)ytt+y=0 .

Find the interval of convergence for the solution of the differential equation yt+3xy=0. y ^ {t} + 3 x y = 0 \text {. }yt+3xy=0.

Find the interval of convergence for the solution of the differential equation 3ytt−xyt−y=03 y ^ { tt } - x y ^ { t } - y = 03ytt−xyt−y=0

Solve the differential equation ytt−9xyt=0y ^ { tt } - 9 x y ^ { t } = 0ytt−9xyt=0

Find the series solution of the differential equation ytt−7xyt=0 given the initial y ^ { tt } - 7 x y ^ { t } = 0 \text { given the initial }ytt−7xyt=0 given the initial conditions y(0)=0y ( 0 ) = 0y(0)=0 and yt(0)=4y ^ { t } ( 0 ) = 4yt(0)=4

Solve the differential equation 144ytt−24yt+y=12(x+ex) by the method of 144 y ^ {tt} - 24 y ^ { t } + y = 12 \left( x + e ^ { x } \right) \text { by the method of }144ytt−24yt+y=12(x+ex) by the method of undetermined coefficients.

Find the first eight terms of the power series representing independent solutions of the differential equation ytt+11x2y=0y ^ { tt } + 11 x ^ { 2 } y = 0ytt+11x2y=0 .

Find the integrating factor of the differential equation (3x3+2y)dx−xdy=0 that is \left( 3 x ^ { 3 } + 2 y \right) d x - x d y = 0 \text { that is }(3x3+2y)dx−xdy=0 that is a function of x or y alone.

Find the particular solution of the differential equation 3yx−8dx+[3ln⁡(x−8)+10y]dy=0\frac { 3 y } { x - 8 } d x + [ 3 \ln ( x - 8 ) + 10 y ] d y = 0x−83y​dx+[3ln(x−8)+10y]dy=0 that satisfies the initial condition y(9)=2y ( 9 ) = 2y(9)=2

Find the particular solution of the differential equation e8x(sin⁡8ydx+cos⁡8ydy)=0e ^ { 8 x } ( \sin 8 y d x + \cos 8 y d y ) = 0e8x(sin8ydx+cos8ydy)=0 that satisfies the initial condition y(0)=πy ( 0 ) = \piy(0)=π .

Graphs and Models

A Preview of Calculus

The Derivative and the Tangent Line Problem

Extrema on an Interval

Antiderivatives and Indefinite Integration

Slope Fields and Eulers Method

Area of a Region Between Two Curves

Basic Integration Rules

Sequences

Conics and Calculus

Vectors in the Plane

Vector-Valued Functions

Introduction to Functions of Several Variables

Iterated Integrals and Area in the Plane

Vector Fields

Filters

Solve the differential equation $10 y ^ { tt } - x y ^ { t } - y = 0$

Solve the differential equation $y ^ { tt } - 6 y ^ { t} + 5 y = 20 x \text { by the method of undetermined }$ coefficients.

Use a power series to solve the differential equation $y ^ { tt } - 25 y = 0$

Find the first six terms of the power series representing independent solutions of the differential equation $\left( x ^ { 2 } + 2 \right) y ^ { tt } + y = 0$ .

Find the interval of convergence for the solution of the differential equation $y ^ {t} + 3 x y = 0 \text {. }$

Find the interval of convergence for the solution of the differential equation $3 y ^ { tt } - x y ^ { t } - y = 0$

Solve the differential equation $y ^ { tt } - 9 x y ^ { t } = 0$

Find the series solution of the differential equation $y ^ { tt } - 7 x y ^ { t } = 0 \text { given the initial }$ conditions $y ( 0 ) = 0$ and $y ^ { t } ( 0 ) = 4$

Solve the differential equation $144 y ^ {tt} - 24 y ^ { t } + y = 12 \left( x + e ^ { x } \right) \text { by the method of }$ undetermined coefficients.

Find the first eight terms of the power series representing independent solutions of the differential equation $y ^ { tt } + 11 x ^ { 2 } y = 0$ .

Find the integrating factor of the differential equation $\left( 3 x ^ { 3 } + 2 y \right) d x - x d y = 0 \text { that is }$ a function of x or y alone.

Find the particular solution of the differential equation $\frac { 3 y } { x - 8 } d x + [ 3 \ln ( x - 8 ) + 10 y ] d y = 0$ that satisfies the initial condition $y ( 9 ) = 2$

Find the particular solution of the differential equation $e ^ { 8 x } ( \sin 8 y d x + \cos 8 y d y ) = 0$ that satisfies the initial condition $y ( 0 ) = \pi$ .