Question 1

Find f by solving the initial value problem. $f ^ { \prime \prime } ( x ) = 8 x ^ { 2 } + 6 x + 4$ ; $f ( - 1 ) = - 7$ , $f ^ { \prime } ( - 1 ) = - 3$

Accepted Answer

A)  $\frac { 8 } { 3 }$ $x ^ { 4 }$ $+ 3$ $x ^ { 3 }$ + 4 $x ^ { 2 }$ $+ \frac { 2 } { 3 }$ x - 10 
B)  $\frac { 2 } { 3 }$ $x ^ { 4 }$ $+$ $x ^ { 3 }$ + 2 $x ^ { 2 }$ $+ \frac { 2 } { 3 }$ x $- 8$ 
C)  $\frac { 2 } { 3 }$ $x ^ { 4 }$ $+$ $x ^ { 3 }$ + 2 $x ^ { 2 }$ $+ \frac { 2 } { 3 }$ x $- 16$ 
D)  $\frac { 8 } { 3 }$ $x ^ { 4 }$ $+ 3$ $x ^ { 3 }$ + 4 $x ^ { 2 }$ $+ \frac { 2 } { 3 }$ x - 5 
A)  $\frac { 8 } { 3 }$ $x ^ { 4 }$ $+ 3$ $x ^ { 3 }$ + 4 $x ^ { 2 }$ $+ \frac { 2 } { 3 }$ x - 10 
B)  $\frac { 2 } { 3 }$ $x ^ { 4 }$ $+$ $x ^ { 3 }$ + 2 $x ^ { 2 }$ $+ \frac { 2 } { 3 }$ x $- 8$ 
C)  $\frac { 2 } { 3 }$ $x ^ { 4 }$ $+$ $x ^ { 3 }$ + 2 $x ^ { 2 }$ $+ \frac { 2 } { 3 }$ x $- 16$ 
D)  $\frac { 8 } { 3 }$ $x ^ { 4 }$ $+ 3$ $x ^ { 3 }$ + 4 $x ^ { 2 }$ $+ \frac { 2 } { 3 }$ x - 5

Question 2

Solve the initial-value problem using the method of undetermined coefficients. $y ^ { \prime \prime } - y ^ { \prime } = x e ^ { x } , y ( 0 ) = 0 , y ^ { \prime } ( 0 ) = 5$

Accepted Answer

The answer of Solve the initial-value problem using the method...

Question 3

Solve the differential equation. $y ^ { \prime \prime } - 3 y ^ { \prime } - 28 y = 0$

Accepted Answer

The answer of Solve the differential equation. \(y ^ {...

Question 4

Solve the differential equation. $y ^ { \prime \prime } + 10 y ^ { \prime } + 61 y = 0$

Accepted Answer

The answer of Solve the differential equation. \(y ^ {...

Question 5

Use power series to solve the differential equation. $$( x - 10 ) y ^ { \prime } + 2 y = 0$$

Accepted Answer

A)  $$y ( x ) = c _ { 0 } \sum _ { n = 0 } ^ { \infty } \frac { x ^ { n } } { 10 ^ { n } }$$ 
B)  $y ( x ) = c _ { 0 } \sum _ { n = 0 } ^ { \infty } \frac { n + 1 } { 10 ^ { n } } x ^ { n }$ 
C)  $y ( x ) = c _ { 0 } \sum _ { n = 0 } ^ { \infty } \frac { ( n + 1 ) } { x ^ { n } }$ 
D)  $$y ( x ) = c _ { 0 } \sum _ { n = 0 } ^ { \infty } ( n + 1 ) x ^ { n }$$ 
E)  $y ( x ) = c _ { 0 } \sum _ { n = 0 } ^ { \infty } \frac { n x ^ { n } } { 10 ^ { n } }$ 
A)  $$y ( x ) = c _ { 0 } \sum _ { n = 0 } ^ { \infty } \frac { x ^ { n } } { 10 ^ { n } }$$ 
B)  $y ( x ) = c _ { 0 } \sum _ { n = 0 } ^ { \infty } \frac { n + 1 } { 10 ^ { n } } x ^ { n }$ 
C)  $y ( x ) = c _ { 0 } \sum _ { n = 0 } ^ { \infty } \frac { ( n + 1 ) } { x ^ { n } }$ 
D)  $$y ( x ) = c _ { 0 } \sum _ { n = 0 } ^ { \infty } ( n + 1 ) x ^ { n }$$ 
E)  $y ( x ) = c _ { 0 } \sum _ { n = 0 } ^ { \infty } \frac { n x ^ { n } } { 10 ^ { n } }$

Question 6

Use power series to solve the differential equation. $y ^ { \prime \prime } - x y ^ { \prime } - y = 0 , y ( 0 ) = 7 , y ^ { \prime } ( 0 ) = 0$

Accepted Answer

A)  $y ( x ) = 7 e ^ { x }$ 
B)  $y ( x ) = 7 e ^ { - \frac { x ^ { 2 } } { 2 } }$ 
C)  $y ( x ) = e ^ { 7 x }$ 
D)  $y ( x ) = c e ^ { \frac { x ^ { 2 } } { 2 } }$ 
E)  $y ( x ) = 7 e ^ { \frac { x ^ { 2 } } { 2 } }$ 
A)  $y ( x ) = 7 e ^ { x }$ 
B)  $y ( x ) = 7 e ^ { - \frac { x ^ { 2 } } { 2 } }$ 
C)  $y ( x ) = e ^ { 7 x }$ 
D)  $y ( x ) = c e ^ { \frac { x ^ { 2 } } { 2 } }$ 
E)  $y ( x ) = 7 e ^ { \frac { x ^ { 2 } } { 2 } }$

Question 7

Solve the differential equation. $y ^ { \prime \prime } - y = \cos x$

Accepted Answer

The answer of Solve the differential equation. \(y ^ {...

Question 8

Use power series to solve the differential equation. $y ^ { \prime \prime } = 25 y$

Accepted Answer

A)  $y ( x ) = c _ { 1 } \cosh 5 x + c _ { 2 } \sinh 5 x$ 
B)  $y ( x ) = c _ { 1 } 5 \cosh x + c _ { 2 } 5 \sinh x + e ^ { x }$ 
C)  $y ( x ) = 5 \cosh x + 5 \sinh x$ 
D)  $y ( x ) = c _ { 1 } \cosh 5 x + c _ { 2 } \sinh 5 x + e ^ { x }$ 
E)  $y ( x ) = c _ { 1 } \cosh 5 x + c _ { 2 } \sinh 5 x + x ^ { 2 }$ 
A)  $y ( x ) = c _ { 1 } \cosh 5 x + c _ { 2 } \sinh 5 x$ 
B)  $y ( x ) = c _ { 1 } 5 \cosh x + c _ { 2 } 5 \sinh x + e ^ { x }$ 
C)  $y ( x ) = 5 \cosh x + 5 \sinh x$ 
D)  $y ( x ) = c _ { 1 } \cosh 5 x + c _ { 2 } \sinh 5 x + e ^ { x }$ 
E)  $y ( x ) = c _ { 1 } \cosh 5 x + c _ { 2 } \sinh 5 x + x ^ { 2 }$

Question 9

Solve the differential equation using the method of undetermined coefficients. $y ^ { \prime \prime } + 6 y ^ { \prime } + 9 y = 2 + x$

Accepted Answer

A)  $y ( x ) = c _ { 1 } e ^ { - 3 x } + c _ { 2 } x e ^ { - 3 x } + \frac { 2 } { 9 } x + \frac { 4 } { 27 }$ 
B)  $y ( x ) = c _ { 1 } e ^ { - 3 x } + c _ { 2 } x e ^ { - 3 x }$ 
C)  $y ( x ) = c _ { 1 } e ^ { - 3 x } + c _ { 2 } x e ^ { - 3 x } + \frac { 2 } { 9 } x$ 
D)  $y ( x ) = c _ { 1 } e ^ { - 3 x } + \frac { 2 } { 9 } x + \frac { 4 } { 27 }$ 
E)  $y ( x ) = c _ { 1 } x e ^ { - 3 x } + c _ { 2 } x ^ { 2 } e ^ { - 3 x } + \frac { 4 } { 27 } x$ 
A)  $y ( x ) = c _ { 1 } e ^ { - 3 x } + c _ { 2 } x e ^ { - 3 x } + \frac { 2 } { 9 } x + \frac { 4 } { 27 }$ 
B)  $y ( x ) = c _ { 1 } e ^ { - 3 x } + c _ { 2 } x e ^ { - 3 x }$ 
C)  $y ( x ) = c _ { 1 } e ^ { - 3 x } + c _ { 2 } x e ^ { - 3 x } + \frac { 2 } { 9 } x$ 
D)  $y ( x ) = c _ { 1 } e ^ { - 3 x } + \frac { 2 } { 9 } x + \frac { 4 } { 27 }$ 
E)  $y ( x ) = c _ { 1 } x e ^ { - 3 x } + c _ { 2 } x ^ { 2 } e ^ { - 3 x } + \frac { 4 } { 27 } x$

Question 10

Solve the boundary-value problem, if possible. $y ^ { \prime \prime } + 10 y ^ { \prime } + 25 y = 0 , y ( 0 ) = 0 , y ( 1 ) = 8$

Accepted Answer

The answer of Solve the boundary-value problem, if possible. \(y...

Question 11

A spring with a mass of 2 kg has damping constant 8 and spring constant 80. Graph the position function of the mass at time t if it starts at the equilibrium position with a velocity of 2 m/s.

Accepted Answer

A)    
B)    
C)    
A)    
B)    
C)

Question 12

Find a trial solution for the method of undetermined coefficients. Do not determine the coefficients. $y ^ { \prime \prime } + 3 y ^ { \prime } + 5 y = x ^ { 4 } e ^ { 9 x }$

Accepted Answer

A)  $y _ { y } ( x ) = \left( A x ^ { 4 } + B x ^ { 2 } + C \right) e ^ { - 9 x }$ 
B)  $y _ { y } ( x ) = \left( A x ^ { 3 } + B x ^ { 2 } + C x + D \right) e ^ { 9 x }$ 
C)  $y _ { y } ( x ) = \left( A x ^ { 4 } + B \right) e ^ { 9 x }$ 
D)  $y _ { y } ( x ) = A x ^ { 9 }$ 
E)  $y _ { y } ( x ) = \left( A x ^ { 4 } + B x ^ { 3 } + C x ^ { 2 } + D x + E \right) e ^ { 9 x }$ 
A)  $y _ { y } ( x ) = \left( A x ^ { 4 } + B x ^ { 2 } + C \right) e ^ { - 9 x }$ 
B)  $y _ { y } ( x ) = \left( A x ^ { 3 } + B x ^ { 2 } + C x + D \right) e ^ { 9 x }$ 
C)  $y _ { y } ( x ) = \left( A x ^ { 4 } + B \right) e ^ { 9 x }$ 
D)  $y _ { y } ( x ) = A x ^ { 9 }$ 
E)  $y _ { y } ( x ) = \left( A x ^ { 4 } + B x ^ { 3 } + C x ^ { 2 } + D x + E \right) e ^ { 9 x }$

Question 13

A spring with a mass of $9$ kg has damping constant 28 and spring constant $195$ . Find the damping constant that would produce critical damping.

Accepted Answer

A)  $c =$ $36 \sqrt { 195 }$ 
B)  $c =$ $2340$ 
C)  $c = 195 \sqrt { 9 }$ 
D)  $c =$ 9 $\sqrt { 195 }$ 
E)  $c =$ $6 \sqrt { 195 }$ 
A)  $c =$ $36 \sqrt { 195 }$ 
B)  $c =$ $2340$ 
C)  $c = 195 \sqrt { 9 }$ 
D)  $c =$ 9 $\sqrt { 195 }$ 
E)  $c =$ $6 \sqrt { 195 }$

Question 14

Find a trial solution for the method of undetermined coefficients. Do not determine the coefficients. $y ^ { \prime \prime } + 2 y ^ { \prime } = 1 + x e ^ { - 4 x }$

Accepted Answer

A)  $\begin{array} { l } 
y _ { p 1 } ( x ) = A \
y _ { p 2 } ( x ) = B x e ^ { 4 x }
\end{array}$ 
B)  $\begin{array} { l } 
y _ { p 1 } ( x ) = A x \
y _ { p 2 } ( x ) = x ( A x + B ) e ^ { - 4 x }
\end{array}$ 
C)  $\begin{array} { l } 
y _ { 71 } ( x ) = A \
y _ { p 2 } ( x ) = x ( A + B ) e ^ { - 4 x }
\end{array}$ 
D)  $\begin{array} { l } 
y _ { p 1 } ( x ) = A x \
y _ { p 2 } ( x ) = x ^ { 2 } ( A x + B ) e ^ { - 4 x }
\end{array}$ 
E)  $\begin{array} { l } 
y _ { p 1 } ( x ) = A \
y _ { p 2 } ( x ) = x ( A + B x ) e ^ { - 4 x }
\end{array}$ 
A)  $\begin{array} { l } 
y _ { p 1 } ( x ) = A \
y _ { p 2 } ( x ) = B x e ^ { 4 x }
\end{array}$ 
B)  $\begin{array} { l } 
y _ { p 1 } ( x ) = A x \
y _ { p 2 } ( x ) = x ( A x + B ) e ^ { - 4 x }
\end{array}$ 
C)  $\begin{array} { l } 
y _ { 71 } ( x ) = A \
y _ { p 2 } ( x ) = x ( A + B ) e ^ { - 4 x }
\end{array}$ 
D)  $\begin{array} { l } 
y _ { p 1 } ( x ) = A x \
y _ { p 2 } ( x ) = x ^ { 2 } ( A x + B ) e ^ { - 4 x }
\end{array}$ 
E)  $\begin{array} { l } 
y _ { p 1 } ( x ) = A \
y _ { p 2 } ( x ) = x ( A + B x ) e ^ { - 4 x }
\end{array}$

Question 15

Solve the differential equation. $y ^ { \prime \prime } - 4 y ^ { \prime } + 13 y = 0$

Accepted Answer

A)  $y ( x ) = c e ^ { 2 x } \cos ( 3 x )$ 
B)  $y ( x ) = e ^ { 2 x } \left( c _ { 1 } \cos ( 3 x ) + c _ { 2 } x \sin ( 3 x ) \right)$ 
C)  $y ( x ) = e ^ { 2 x } \left( c _ { 1 } \cos ( 3 x ) + c _ { 2 } \sin ( 3 x ) \right)$ 
D)  $y ( x ) = e ^ { 3 x } \left( c _ { 1 } \cos ( 2 x ) + c _ { 2 } \sin ( 2 x ) \right)$ 
E)  $y ( x ) = e ^ { 2 x } \left( c _ { 1 } \cos ( 2 x ) + c _ { 2 } x \sin ( 2 x ) \right)$ 
A)  $y ( x ) = c e ^ { 2 x } \cos ( 3 x )$ 
B)  $y ( x ) = e ^ { 2 x } \left( c _ { 1 } \cos ( 3 x ) + c _ { 2 } x \sin ( 3 x ) \right)$ 
C)  $y ( x ) = e ^ { 2 x } \left( c _ { 1 } \cos ( 3 x ) + c _ { 2 } \sin ( 3 x ) \right)$ 
D)  $y ( x ) = e ^ { 3 x } \left( c _ { 1 } \cos ( 2 x ) + c _ { 2 } \sin ( 2 x ) \right)$ 
E)  $y ( x ) = e ^ { 2 x } \left( c _ { 1 } \cos ( 2 x ) + c _ { 2 } x \sin ( 2 x ) \right)$

Question 16

Solve the initial-value problem using the method of undetermined coefficients. $y ^ { \prime \prime } + y ^ { \prime } - 2 y = x + \sin x , y ( 0 ) = \frac { 19 } { 20 } , y ^ { \prime } ( 0 ) = 0$

Accepted Answer

The answer of Solve the initial-value problem using the method...

Question 17

Solve the differential equation. $7 y ^ { \prime \prime } = 4 y ^ { \prime }$

Accepted Answer

The answer of Solve the differential equation. \(7 y ^...

Question 18

Solve the differential equation using the method of variation of parameters. $y ^ { \prime \prime } - 25 y ^ { \prime } = \frac { 1 } { x }$

Accepted Answer

The answer of Solve the differential equation using the method...

Question 19

Solve the differential equation. $\frac { d ^ { 2 } y } { d t ^ { 2 } } + \frac { d y } { d t } + 3 y = 0$

Accepted Answer

A)  $y ( t ) = t e ^ { \frac { t } { 2 } } \left[ c _ { 1 } \cos \left( \frac { \sqrt { 11 } t } { 2 } \right) + c _ { 2 } \sin \left( \frac { \sqrt { 11 } t } { 2 } \right) \right]$ 
B)  $y ( t ) = e ^ { \frac { \sqrt { 2 } } { 2 } } \left[ c _ { 1 } \cos \left( \frac { \sqrt { t } } { 2 } \right) + c _ { 2 } \sin \left( \frac { \sqrt { t } } { 2 } \right) \right]$ 
C)  $y ( t ) = e ^ { - \frac { t } { 2 } } \left[ c _ { 1 } \cos \left( \frac { \sqrt { 11 } t } { 2 } \right) + c _ { 2 } \sin \left( \frac { \sqrt { 11 } t } { 2 } \right) \right]$ 
D)  $y ( t ) = t ^ { 2 } e ^ { - \frac { t } { 2 } } \left[ c _ { 1 } \cos \left( \sqrt { \frac { 11 t } { 2 } } \right) + c _ { 2 } \sin \left( \sqrt { \frac { 11 t } { 2 } } \right) \right]$ 
E)  $y ( t ) = c e ^ { - \frac { t } { 2 } } \cos \left( \sqrt { \frac { 11 t } { 2 } } \right)$ 
A)  $y ( t ) = t e ^ { \frac { t } { 2 } } \left[ c _ { 1 } \cos \left( \frac { \sqrt { 11 } t } { 2 } \right) + c _ { 2 } \sin \left( \frac { \sqrt { 11 } t } { 2 } \right) \right]$ 
B)  $y ( t ) = e ^ { \frac { \sqrt { 2 } } { 2 } } \left[ c _ { 1 } \cos \left( \frac { \sqrt { t } } { 2 } \right) + c _ { 2 } \sin \left( \frac { \sqrt { t } } { 2 } \right) \right]$ 
C)  $y ( t ) = e ^ { - \frac { t } { 2 } } \left[ c _ { 1 } \cos \left( \frac { \sqrt { 11 } t } { 2 } \right) + c _ { 2 } \sin \left( \frac { \sqrt { 11 } t } { 2 } \right) \right]$ 
D)  $y ( t ) = t ^ { 2 } e ^ { - \frac { t } { 2 } } \left[ c _ { 1 } \cos \left( \sqrt { \frac { 11 t } { 2 } } \right) + c _ { 2 } \sin \left( \sqrt { \frac { 11 t } { 2 } } \right) \right]$ 
E)  $y ( t ) = c e ^ { - \frac { t } { 2 } } \cos \left( \sqrt { \frac { 11 t } { 2 } } \right)$

Question 20

Solve the differential equation. $y ^ { \prime \prime } + 5 y ^ { \prime } - 6 y = 0$

Accepted Answer

A)  $y = C _ { 1 } e ^ { x } + C _ { 2 } e ^ { - 7 x }$ 
B)  $y = C _ { 1 } e ^ { - 6 x } + C _ { 2 } e ^ { - 6 x }$ 
C)  $y = \frac { C _ { 1 } e ^ { x } } { 3 } + \frac { C _ { 2 } e ^ { - 6 x } } { 2 }$ 
D)  $y = C _ { 1 } e ^ { x } + C _ { 2 } e ^ { - 6 x }$ 
E)  None of these 
A)  $y = C _ { 1 } e ^ { x } + C _ { 2 } e ^ { - 7 x }$ 
B)  $y = C _ { 1 } e ^ { - 6 x } + C _ { 2 } e ^ { - 6 x }$ 
C)  $y = \frac { C _ { 1 } e ^ { x } } { 3 } + \frac { C _ { 2 } e ^ { - 6 x } } { 2 }$ 
D)  $y = C _ { 1 } e ^ { x } + C _ { 2 } e ^ { - 6 x }$ 
E)  None of these

Find f by solving the initial value problem. $f ^ { \prime \prime } ( x ) = 8 x ^ { 2 } + 6 x + 4$ ; $f ( - 1 ) = - 7$ , $f ^ { \prime } ( - 1 ) = - 3$

Solve the initial-value problem using the method of undetermined coefficients. $y ^ { \prime \prime } - y ^ { \prime } = x e ^ { x } , y ( 0 ) = 0 , y ^ { \prime } ( 0 ) = 5$

Solve the differential equation. $y ^ { \prime \prime } - 3 y ^ { \prime } - 28 y = 0$

Solve the differential equation. $y ^ { \prime \prime } + 10 y ^ { \prime } + 61 y = 0$

Use power series to solve the differential equation. $( x - 10 ) y ^ { \prime } + 2 y = 0$

Use power series to solve the differential equation. $y ^ { \prime \prime } - x y ^ { \prime } - y = 0 , y ( 0 ) = 7 , y ^ { \prime } ( 0 ) = 0$

Solve the differential equation. $y ^ { \prime \prime } - y = \cos x$

Use power series to solve the differential equation. $y ^ { \prime \prime } = 25 y$

Solve the differential equation using the method of undetermined coefficients. $y ^ { \prime \prime } + 6 y ^ { \prime } + 9 y = 2 + x$

Solve the boundary-value problem, if possible. $y ^ { \prime \prime } + 10 y ^ { \prime } + 25 y = 0 , y ( 0 ) = 0 , y ( 1 ) = 8$

A spring with a mass of 2 kg has damping constant 8 and spring constant 80. Graph the position function of the mass at time t if it starts at the equilibrium position with a velocity of 2 m/s.

Find a trial solution for the method of undetermined coefficients. Do not determine the coefficients. $y ^ { \prime \prime } + 3 y ^ { \prime } + 5 y = x ^ { 4 } e ^ { 9 x }$

A spring with a mass of $9$ kg has damping constant 28 and spring constant $195$ . Find the damping constant that would produce critical damping.

Find a trial solution for the method of undetermined coefficients. Do not determine the coefficients. $y ^ { \prime \prime } + 2 y ^ { \prime } = 1 + x e ^ { - 4 x }$

Solve the differential equation. $y ^ { \prime \prime } - 4 y ^ { \prime } + 13 y = 0$

Solve the initial-value problem using the method of undetermined coefficients. $y ^ { \prime \prime } + y ^ { \prime } - 2 y = x + \sin x , y ( 0 ) = \frac { 19 } { 20 } , y ^ { \prime } ( 0 ) = 0$

Solve the differential equation. $7 y ^ { \prime \prime } = 4 y ^ { \prime }$

Solve the differential equation using the method of variation of parameters. $y ^ { \prime \prime } - 25 y ^ { \prime } = \frac { 1 } { x }$

Solve the differential equation. $\frac { d ^ { 2 } y } { d t ^ { 2 } } + \frac { d y } { d t } + 3 y = 0$

Solve the differential equation. $y ^ { \prime \prime } + 5 y ^ { \prime } - 6 y = 0$

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Applications of Integration

Inverse Functions

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Final Exam

Filters

Exam 17: Second-Order Differential Equations

Find f by solving the initial value problem. f′′(x)=8x2+6x+4f ^ { \prime \prime } ( x ) = 8 x ^ { 2 } + 6 x + 4f′′(x)=8x2+6x+4 ; f(−1)=−7f ( - 1 ) = - 7f(−1)=−7 , f′(−1)=−3f ^ { \prime } ( - 1 ) = - 3f′(−1)=−3

Solve the initial-value problem using the method of undetermined coefficients. y′′−y′=xex,y(0)=0,y′(0)=5y ^ { \prime \prime } - y ^ { \prime } = x e ^ { x } , y ( 0 ) = 0 , y ^ { \prime } ( 0 ) = 5y′′−y′=xex,y(0)=0,y′(0)=5

Solve the differential equation. y′′−3y′−28y=0y ^ { \prime \prime } - 3 y ^ { \prime } - 28 y = 0y′′−3y′−28y=0

Solve the differential equation. y′′+10y′+61y=0y ^ { \prime \prime } + 10 y ^ { \prime } + 61 y = 0y′′+10y′+61y=0

Use power series to solve the differential equation. (x−10)y′+2y=0( x - 10 ) y ^ { \prime } + 2 y = 0(x−10)y′+2y=0

Use power series to solve the differential equation. y′′−xy′−y=0,y(0)=7,y′(0)=0y ^ { \prime \prime } - x y ^ { \prime } - y = 0 , y ( 0 ) = 7 , y ^ { \prime } ( 0 ) = 0y′′−xy′−y=0,y(0)=7,y′(0)=0

Solve the differential equation. y′′−y=cos⁡xy ^ { \prime \prime } - y = \cos xy′′−y=cosx

Use power series to solve the differential equation. y′′=25yy ^ { \prime \prime } = 25 yy′′=25y

Solve the differential equation using the method of undetermined coefficients. y′′+6y′+9y=2+xy ^ { \prime \prime } + 6 y ^ { \prime } + 9 y = 2 + xy′′+6y′+9y=2+x

Solve the boundary-value problem, if possible. y′′+10y′+25y=0,y(0)=0,y(1)=8y ^ { \prime \prime } + 10 y ^ { \prime } + 25 y = 0 , y ( 0 ) = 0 , y ( 1 ) = 8y′′+10y′+25y=0,y(0)=0,y(1)=8

A spring with a mass of 2 kg has damping constant 8 and spring constant 80. Graph the position function of the mass at time t if it starts at the equilibrium position with a velocity of 2 m/s.

Find a trial solution for the method of undetermined coefficients. Do not determine the coefficients. y′′+3y′+5y=x4e9xy ^ { \prime \prime } + 3 y ^ { \prime } + 5 y = x ^ { 4 } e ^ { 9 x }y′′+3y′+5y=x4e9x

A spring with a mass of 999 kg has damping constant 28 and spring constant 195195195 . Find the damping constant that would produce critical damping.

Find a trial solution for the method of undetermined coefficients. Do not determine the coefficients. y′′+2y′=1+xe−4xy ^ { \prime \prime } + 2 y ^ { \prime } = 1 + x e ^ { - 4 x }y′′+2y′=1+xe−4x

Solve the differential equation. y′′−4y′+13y=0y ^ { \prime \prime } - 4 y ^ { \prime } + 13 y = 0y′′−4y′+13y=0

Solve the initial-value problem using the method of undetermined coefficients. y′′+y′−2y=x+sin⁡x,y(0)=1920,y′(0)=0y ^ { \prime \prime } + y ^ { \prime } - 2 y = x + \sin x , y ( 0 ) = \frac { 19 } { 20 } , y ^ { \prime } ( 0 ) = 0y′′+y′−2y=x+sinx,y(0)=2019​,y′(0)=0

Solve the differential equation. 7y′′=4y′7 y ^ { \prime \prime } = 4 y ^ { \prime }7y′′=4y′

Solve the differential equation using the method of variation of parameters. y′′−25y′=1xy ^ { \prime \prime } - 25 y ^ { \prime } = \frac { 1 } { x }y′′−25y′=x1​

Solve the differential equation. d2ydt2+dydt+3y=0\frac { d ^ { 2 } y } { d t ^ { 2 } } + \frac { d y } { d t } + 3 y = 0dt2d2y​+dtdy​+3y=0

Solve the differential equation. y′′+5y′−6y=0y ^ { \prime \prime } + 5 y ^ { \prime } - 6 y = 0y′′+5y′−6y=0

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Applications of Integration

Inverse Functions

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Final Exam

Filters

Find f by solving the initial value problem. $f ^ { \prime \prime } ( x ) = 8 x ^ { 2 } + 6 x + 4$ ; $f ( - 1 ) = - 7$ , $f ^ { \prime } ( - 1 ) = - 3$

Solve the initial-value problem using the method of undetermined coefficients. $y ^ { \prime \prime } - y ^ { \prime } = x e ^ { x } , y ( 0 ) = 0 , y ^ { \prime } ( 0 ) = 5$

Solve the differential equation. $y ^ { \prime \prime } - 3 y ^ { \prime } - 28 y = 0$

Solve the differential equation. $y ^ { \prime \prime } + 10 y ^ { \prime } + 61 y = 0$

Use power series to solve the differential equation. $( x - 10 ) y ^ { \prime } + 2 y = 0$

Use power series to solve the differential equation. $y ^ { \prime \prime } - x y ^ { \prime } - y = 0 , y ( 0 ) = 7 , y ^ { \prime } ( 0 ) = 0$

Solve the differential equation. $y ^ { \prime \prime } - y = \cos x$

Use power series to solve the differential equation. $y ^ { \prime \prime } = 25 y$

Solve the differential equation using the method of undetermined coefficients. $y ^ { \prime \prime } + 6 y ^ { \prime } + 9 y = 2 + x$

Solve the boundary-value problem, if possible. $y ^ { \prime \prime } + 10 y ^ { \prime } + 25 y = 0 , y ( 0 ) = 0 , y ( 1 ) = 8$

Find a trial solution for the method of undetermined coefficients. Do not determine the coefficients. $y ^ { \prime \prime } + 3 y ^ { \prime } + 5 y = x ^ { 4 } e ^ { 9 x }$

A spring with a mass of $9$ kg has damping constant 28 and spring constant $195$ . Find the damping constant that would produce critical damping.

Find a trial solution for the method of undetermined coefficients. Do not determine the coefficients. $y ^ { \prime \prime } + 2 y ^ { \prime } = 1 + x e ^ { - 4 x }$

Solve the differential equation. $y ^ { \prime \prime } - 4 y ^ { \prime } + 13 y = 0$

Solve the initial-value problem using the method of undetermined coefficients. $y ^ { \prime \prime } + y ^ { \prime } - 2 y = x + \sin x , y ( 0 ) = \frac { 19 } { 20 } , y ^ { \prime } ( 0 ) = 0$

Solve the differential equation. $7 y ^ { \prime \prime } = 4 y ^ { \prime }$

Solve the differential equation using the method of variation of parameters. $y ^ { \prime \prime } - 25 y ^ { \prime } = \frac { 1 } { x }$

Solve the differential equation. $\frac { d ^ { 2 } y } { d t ^ { 2 } } + \frac { d y } { d t } + 3 y = 0$

Solve the differential equation. $y ^ { \prime \prime } + 5 y ^ { \prime } - 6 y = 0$