Exam 10: Differential Equations

Let t represent the number of hours that a packing machine is operated and y(t) represent the probability that the machine breaks down at least once during the t hours of operation. It has been observed that the rate of increase of the probability of a breakdown is proportional to the probability of not having a breakdown. Find a differential equation describing this situation.

The following is a polygonal path obtained from Euler's method with n = 4 to approximate a solution f(t) of a differential equation. Indicate whether the following statements are true or false: - $f ^ { \prime }$ $\left( \frac { 1 } { 2 } \right)$ ≈ 1

Below is a sketch of f(x) = (x - 1) $e ^ { x }$ . On a ty-coordinate system, sketch the solutions to the differential equation $y ^ { \prime } = ( y - 1 ) e y$ corresponding to the initial conditions $y ( 0 ) = 2$ , $y ( 0 ) = \frac { 1 } { 2 }$ , and $y ( 0 ) = - \frac { 1 } { 2 }$ . Does the following graph represent this situation?

One or more initial conditions are given for the differential equation. Use the qualitative theory of autonomous differential equations to sketch the graphs of the corresponding solution. Include a yz-graph as well as a ty-graph. $y ^ { \prime } = y ^ { 3 } - 3 y ^ { 2 } ; y ( 0 ) = - 1.5 ; y ( 0 ) = 2.5$ Do these graphs represent the situation?

Solve the differential equation with the given initial condition. - $y ^ { \prime } = y + 1 , y ( 0 ) = 1$

Use Euler's method with n = 5 to approximate f(1) if y = f(t) satisfies the differential equation $y ^ { \prime } = y$ $y ( 0 ) = 1$ Compare this answer with the exact value of f(1). Is the following the correct answer? Euler's method: f(1) ≈ 2.488; Actual value: f(t) = e ≈ 2.718

Solve the initial value problem using an integrating factor. - $y ^ { \prime }$ + y = 3; y(0) = 0.

Write a differential equation that expresses the following description of a rate: When ice cream is removed from the freezer, it warms up at a rate proportional to the difference between the temperature of the ice cream and the room temperature of 76°. (Use y for the temperature of the ice cream, t for the time, and k for an unknown constant.)

Consider the differential equation $y ^ { \prime }$ = g(y) where g(y) is the function whose graph is shown below: Indicate whether the following statements are true or false. -If the initial value of y(0) is greater than 6, then the corresponding solution will be an increasing function.

The birth rate in a certain city is 2% per year and the death rate is 2.5% per year. Also, there is a net movement of population into the city at the rate of 4000 people per year. Let N = f(t) be the city's population at time t. Write the differential equation satisfied by f(t). Does this equation accurately represent this situation: $y ^ { \prime } = 4000 - 0.005 y ?$

Use Euler's method with n = 3 to approximate the solution f(t) to $y ^ { \prime }$ = 9t + $y ^ { 2 }$ , $y ( 0 ) = 0$ . Estimate f(1). Enter just a reduced fraction of form $\frac { a } { b }$ .

Which of the following functions solves the differential equation: $y ^ { \prime } = - 4 y$ ?

The following is a polygonal path obtained from Euler's method with n = 4 to approximate a solution f(t) of a differential equation. Indicate whether the following statements are true or false: -f(1) ≈ $\frac { 5 } { 2 }$

Given the differential equation with the given initial condition: $\frac { d y } { d t } = y ^ { 2 } \ln t ; \quad y ( 1 ) = \frac { 1 } { 3 }$ is this the solution $y = \frac { 1 } { 2 - t \ln t + t } ?$

A man opens a savings account that earns interest at an annual rate of 6% compounded continuously. He plans to make continuous withdrawals at a rate of $300 per year. What will happen if his initial deposit is $5000? [Hint: Let f(x) be the savings account balance at time t, and determine the differential equation satisfied by f(t).]

Given the differential equation: $y ^ { \prime } = \frac { t \sin t ^ { 2 } } { y }$ is this the solution $y = \pm \sqrt { C - \cos t ^ { 2 } } ?$

Solve the equation using an integrating factor. - $y ^ { \prime } - 3 y = 6$ , t > 0

Consider the differential equation $y ^ { \prime }$ = g(y) where g(y) is the function whose graph is shown below: Indicate whether the following statements are true or false. -For what y value(s) does a solution of $y ^ { \prime }$ = $y ^ { 2 }$ - 3y + 2 have inflection points?

An investment earns 25% interest per year. Every year $10,000 is withdrawn in order to pay dividends to the investors. Set up a differential equation satisfied by f(t), the amount of money invested at time t. Does this equation accurately describe this situation: $y ^ { \prime } = 0.25 y - 10,000 ?$

The following is a polygonal path obtained from Euler's method with n = 4 to approximate a solution f(t) of a differential equation. Indicate whether the following statements are true or false: - $f ^ { \prime }$ $\left( \frac { 5 } { 2 } \right)$ = 1