Deck 8: Differential Equations

Question 1

A desert preserve in the southwest is known for its populations of coyotes and road runners.The growth rate for each population can be modeled by this pair of differential equations: $\begin{array} { l } \frac { d R } { d t } = 2 R - 0.2 R C \\\frac { d C } { d t } = - 1.4 C + 0.02 R C\end{array}$ where C is the number of coyotes and R is the number of road runners.Find the equilibrium populations for this model.(Hint: These are the populations for which $\frac { d R } { d t } = \frac { d C } { d t } = 0$ ).

A)60 coyotes,15 road runners
B)10 coyotes,70 road runners
C)15 coyotes,60 road runners
D)70 coyotes,10 road runners

Answer 1

10 coyotes,70 road runners

Question 2

Find the general solution of $\frac { d y } { d x } = 7 y$ .

A)y = ln x + C
B) $y = e ^ { 7 x } + C$
C)y = 7x + C
D) $y = C e ^ { 7 x }$

Answer 2

$y = C e ^ { 7 x }$

Question 3

Solve the given first-order linear initial value problem. $- y _ { n + 1 } = y _ { n } ; y _ { 0 } = 1$

A) $y _ { n } = 1$
B) $y _ { n } = ( - 1 ) ^ { n + 1 }$
C) $y _ { n } = ( - 1 ) ^ { n }$
D) $y _ { n } = ( - 1 ) ^ { n } + 1$

Answer 3

$y _ { n } = ( - 1 ) ^ { n }$

Question 4

Find the general solution of the given first-order linear differential equation. $\frac { d y } { d x } + \frac { 9 y } { x } = 4 x$

A) $y = \frac { x ^ { 2 } } { 11 } + \frac { C } { x ^ { 9 } }$
B) $y = \frac { 4 x ^ { 2 } } { 11 } + \frac { C } { x ^ { 11 } }$
C) $y = \frac { 4 x ^ { 2 } } { 11 } + \frac { C } { x ^ { 9 } }$
D) $y = \frac { 11 x ^ { 2 } } { 4 } + \frac { C } { x ^ { 9 } }$

Question 5

The price of a certain house is currently $260,000.Suppose it is estimated that after t months,the price $$p ( t )$$ will be increasing at the rate of $0.01 p ( t ) + 1,000 t$ dollars per month.
In 7 months from now,to the nearest whole dollar,the price of the house will be $303,934.

Question 6

Use Euler's method with the step size $h = 0.2$ to estimate the solution $y ( 1 )$ of the given initial value problem.Round your answer to two decimal places. $y ^ { \prime } = \frac { x - y } { 4 x + y } ; y ( 0 ) = 4$

A)1.64
B)4.92
C)6.56
D)3.28

Question 7

Find the general solution of $\frac { d y } { d x } = x ^ { 3 } + 9$ .

A)y = 9x + C
B) $y = 4 x ^ { 4 } + 9 x + C$
C) $y = 3 x ^ { 2 } + C$
D) $y = \frac { x ^ { 4 } } { 4 } + 9 x + C$

Question 8

Find the general solution of the given first-order linear differential equation. $\frac { d y } { d x } + \frac { y } { 20 x } = \sqrt [ 20 ] { x ^ { 19 } } e ^ { x }$

A) $y = \frac { e ^ { x } } { x ^ { 1 / 20 } } + C$
B) $y = \frac { e ^ { x } ( x - 1 ) + C } { x ^ { 20 } }$
C) $y = x ^ { 1 / 20 } e ^ { x } ( x - 1 ) + C$
D) $y = \frac { e ^ { x } ( x - 1 ) + C } { x ^ { 1 / 20 } }$

Question 9

Find the general solution of $\frac { d y } { d x } = \frac { 19 x } { y ^ { 2 } }$ .

A) $y = C$
B) $y = \sqrt [ 3 ] { \frac { 57 x ^ { 2 } } { 2 } + C }$
C) $y = \sqrt [ 3 ] { \frac { ( 38 x ) ^ { 2 } } { 3 } + C }$
D) $y = \frac { 19 x ^ { 2 } } { 2 } + C$

Question 10

The first five terms of the initial value problem $y _ { n } = y _ { n - 1 } ^ { 2 } ; y _ { 0 } = 2$ are 2,4,8,16,and 32.

Question 11

Find the particular solution of the given differential equation that satisfies the indicated condition: $\frac { d y } { d x } = y ^ { 2 } \sqrt { 6 - x }$ ; y = 1 when x = 6.

A) $y = \frac { 2 ( 6 - x ) ^ { 3 / 2 } - 3 } { 3 }$
B) $y = \frac { 3 } { 2 ( 6 + x ) ^ { 3 / 2 } - 3 }$
C) $y = \frac { 3 } { 2 ( 6 - x ) ^ { 3 / 2 } - 3 }$
D) $y = \frac { 3 } { 2 ( 6 - x ) ^ { 3 / 2 } + 3 }$

Question 12

Write a differential equation describing the given situation.Define all variables you introduce.(Do not try to solve the differential equation at this time.)An investment grows at a rate of 3% of its size.

A)Let Q denote the investment and let t denote time; $\frac { d Q } { d t } = 0.03 Q$
B)Let Q denote the investment and let t denote time; $\frac { d Q } { d t } = 0.03 t$
C)Let Q denote the investment and let t denote time; $\frac { d Q } { d t } = 0.3 Q$
D)Let Q denote the investment and let t denote time; $\frac { d Q } { d t } = 3 Q$

Question 13

Find an equation for the orthogonal trajectories of the given family of curves. $8 x ^ { 2 } + y = C$

A) $y - \ln x ^ { 16 } = C$
B) $y - x e ^ { 1 / 16 } = C$
C) $y - \ln x ^ { 1 / 6 } = C$
D) $\ln y - x ^ { 1 / 6 } = C$

Question 14

Find the particular solution of $\frac { d y } { d x } = \frac { x ^ { 2 } y ^ { 3 } } { 5 }$ ,given y = 6 when x = 0.

A) $y = \frac { 2 } { \left( x ^ { 3 } - 6 \right) ^ { 2 } }$
B) $y = 4 x ^ { 6 } + 6$
C) $y = \sqrt { \frac { 180 } { 5 - 24 x ^ { 3 } } }$
D) $y = \frac { x ^ { 3 } } { 3 } + 6$

Question 15

Find the particular solution of the given differential equation that satisfies the given condition. $\frac { d y } { d x } - y = 2 x ^ { 2 } ; y = 1 \text { when } x = 3$

A) $y = \frac { 35 e ^ { x - 3 } } { x ^ { 2 } + 2 x + 2 }$
B) $y = 35 e ^ { x - 3 } - 2 \left( x ^ { 2 } + 2 x + 2 \right)$
C) $y = 35 e ^ { x } + 2 \left( x ^ { 2 } + 2 x + 2 \right)$
D) $y = C e ^ { x - 3 } - 2 \left( x ^ { 2 } + 2 x + 2 \right)$

Question 16

Find the general solution of $\frac { d y } { d x } = e ^ { 7 x }$ .

A) $y = \frac { e ^ { 7 x } } { 7 } + C$
B) $y = \frac { ( 7 x ) ^ { e } } { C }$
C) $y = 7 x e ^ { 7 x } + C$
D) $y = \frac { C e ^ { 7 x } } { 7 }$

Question 17

A dead body is discovered at 8:00 A.M.on Tuesday in a basement where the air temperature is $60 ^ { \circ } \mathrm { F }$ The temperature of the body at the time of discovery is $72 ^ { \circ } \mathrm { F }$ and 20 minutes later,the temperature is $71 ^ { \circ } \mathrm { F }$ The time of death was 6:00 A.M.on Tuesday.

Question 18

Find constants A and B so that the given expression $y _ { n }$ satisfies the specified difference equation. $n ^ { 2 } y _ { n } + n y _ { n - 1 } = 4 n - 2 n ^ { 3 } ; y _ { n } = A n + B$

A) $A = 2 , B = 2$
B) $A = - 2 , B = - 2$
C) $A = 2 , B = - 2$
D) $A = - 2 , B = 2$

Question 19

Find the particular solution of $\frac { d y } { d x } = x ^ { 2 }$ ,given y = 18 when x = 1.

A) $y = \frac { x ^ { 3 } } { 3 } - 53$
B) $y = \frac { x ^ { 3 } } { 3 } - \frac { 53 } { 3 }$
C) $y = \frac { x ^ { 3 } } { 3 } + 18$
D) $y = \frac { x ^ { 3 } } { 3 } + \frac { 53 } { 3 }$

Question 20

The intensity of light $I ( d )$ at a depth d below the surface of a body of water changes at a rate proportional to I.If the intensity at a depth of $d = 2.4$ feet is half of the surface intensity $I _ { 0 }$ to the nearest tenth of a foot,at what depth is the intensity $15 \%$ of $I _ { 0 } ?$

A)6.6 feet
B)9.9 feet
C)13.2 feet
D)7.9 feet