Exam 9: Differential Equations

We modeled populations of aphids and ladybugs with a Lotka-Volterra system. Suppose we modify those equations as follows: $\frac { d A } { d t } = 2 A ( 1 - 0.0005 A ) - 0.01 A L$ , $\frac { d L } { d t } = - 0.6 L + 0.0005 A L$

Suppose that a population develops according to the logistic equation $\frac { d P } { d t } = 0.06 P - 0.0006 P ^ { 2 }$ , where t is measured in weeks. What is the carrying capacity?

In the circuit shown in Figure, a generator supplies a voltage of $E ( t ) = 20 \sin 60 t$ volts, the inductance is 2 H, the resistance is 40 Ω , and $I ( 0 ) = 2 A$ . Find the current 0.2 s after the switch is closed. Round your answer to two decimal places.

For what nonzero values of k does the function $y = A \sin k t + B \cos k t$ satisfy the differential equation $y ^ { \prime \prime } + 100 y = 0$ for all values of A and B?

Determine whether the differential equation is linear. $y ^ { t } + y \ln x = x ^ { 2 } y ^ { 2 }$

Solve the initial-value problem. $\frac { d u } { d t } = \frac { 2 t + \sec ^ { 2 } t } { 2 u } , u ( 0 ) = 4$

Which of the following functions is a solution of the differential equation? $y ^ { \prime \prime } + 16 y ^ { \prime } + 64 y = 0$

A common inhabitant of human intestines is the bacterium Escherichia coli. A cell of this bacterium in a nutrient-broth medium divides into two cells every $20 \text { minutes }$ . The initial population of a culture is $25$ cells. Find the number of cells after $6$ hours.

Find the orthogonal trajectories of the family of curves. $y = k x ^ { 9 }$

A sum of $\$ 5,000$ is invested at $35 \%$ interest. If $A ( t )$ is the amount of the investment at time t for the case of continuous compounding, write a differential equation and an initial condition satisfied by $A ( t )$ .

Solve the initial-value problem. $x y ^ { \prime } - y = x \ln x , \quad y ( 1 ) = 4$

Solve the differential equation. $y ^ { \prime } = \frac { 7 x ^ { 6 } y } { \ln y }$

Kirchhoff's Law gives us the derivative equation $Q ^ { \prime } = 12 - 4 Q$ . If $Q ( 0 ) = 0$ , use Euler's method with step size 0.1 to estimate $Q$ after 0.3 second.

The population of the world was about 5.3 billion in 1990. Birth rates in the 1990s range from 35 to 40 million per year and death rates range from 15 to 20 million per year. Let's assume that the carrying capacity for world population is 100 billion. Use the logistic model to predict the world population in the 2,450 year. Calculate your answer in billions to one decimal place. (Because the initial population is small compared to the carrying capacity, you can take k to be an estimate of the initial relative growth rate.)

Solve the initial-value problem. $x y ^ { \prime } = y + x ^ { 2 } \sin x , y ( 3 \pi ) = 0$

A curve passes through the point $( 8,2 )$ and has the property that the slope of the curve at every point P is $3$ times the y-coordinate P. What is the equation of the curve?

Solve the differential equation. $3 y y ^ { \prime } = 7 x$

Determine whether the differential equation is linear. $y ^ { \prime } + 3 x ^ { 2 } y = 6 x ^ { 2 }$

A population is modeled by the differential equation $\frac { d p } { d t } = 1.8 P \left( 1 - \frac { P } { 5120 } \right)$ . For what values of P is the population decreasing?

One model for the spread of a rumor is that the rate of spread is proportional to the product of the fraction of the population who have heard the rumor and the fraction who have not heard the rumor. Let's assume that the constant of proportionality is $k = 0.03$ . Write a differential equation that is satisfied by y.