Problem 5
For the following problems, consider the logistic equation in the form \(P^{\prime}=C P-P^{2} .\) Draw the directional field and find the stability of the equilibria.\(C=-3\)
Problem 6
Solve the initial-value problem \(y^{\prime}-2 y=4 x+3 \quad y(0)=-2\).
Problem 6
A cake is removed from the oven after baking thoroughly, and the temperature of the oven is \(450^{\circ} \mathrm{F}\). The temperature of the kitchen is \(70^{\circ} \mathrm{F}\), and after 10 minutes the temperature of the cake is \(430^{\circ} \mathrm{F}\). a. Write the appropriate initial-value problem to describe this situation. b. Solve the initial-value problem for \(T(t)\). c. How long will it take until the temperature of the cake is within \(5^{\circ} \mathrm{F}\) of room temperature?
Problem 6
Verifying a Solution to an Initial-Value Problem Verify that the function \(y=2 e^{-2 t}+e^{t}\) is a solution to the initial- value problem $$ y^{\prime}+2 y=3 e^{t}, \quad y(0)=3 $$
Problem 6
For the following problems, consider the logistic equation in the form \(P^{\prime}=C P-P^{2} .\) Draw the directional field and find the stability of the equilibria.Solve the logistic equation for \(C=10\) and an initial condition of \(P(0)=2\).
Problem 7
Create a direction field for the differential equation \(y^{\prime}=(y-3)^{2}\left(y^{2}+y-2\right)\) and identify any equilibrium solutions. Classify each of the equilibrium solutions as stable, unstable, or semi-stable.
Problem 7
For the following problems, consider the logistic equation in the form \(P^{\prime}=C P-P^{2} .\) Draw the directional field and find the stability of the equilibria.Solve the logistic equation for \(C=-10\) and an initial condition of \(P(0)=2\).
Problem 7
Solve the following initial-value problems with the initial condition \(y_{0}=0\) and graph the solution.\(\frac{d y}{d t}=y+1\)
Problem 7
Verify that \(y=3 e^{2 t}+4 \sin t\) is a solution to the initial-value problem $$ y^{\prime}-2 y=4 \cos t-8 \sin t, \quad y(0)=3 $$ Hint First verify that \(y\) solves the differential equation. Then check the initial value.
Problem 7
A racquetball is hit straight upward with an initial velocity of \(2 \mathrm{~m} / \mathrm{s}\). The mass of a racquetball is approximately \(0.0427 \mathrm{~kg}\) Air resistance acts on the ball with a force numerically equal to \(0.5 v\), where \(v\) represents the velocity of the ball at time \(t\). a. Find the velocity of the ball as a function of time. b. How long does it take for the ball to reach its maximum height? c. If the ball is hit from an initial height of 1 meter, how high will it reach?
