Chapter 9

If the solution to \(y^{\prime}=5 x-4 y\) passes through the point \((4,3),\) what is the slope of the solution at that point?

Solve and check each first-order linear differential equation. $$ y^{\prime}+2 y=8 $$

Verify that \(y(t)=c e^{\text {at }}\) solves the differential equation for unlimited growth, \(y^{\prime}=a y\), with initial condition \(y(0)=c\)

Verify that the function \(y\) satisfies the given differential equation. $$ \begin{array}{l} y=e^{2 x}-3 e^{x}+2 \\ y^{\prime \prime}-3 y^{\prime}+2 y=4 \end{array} $$

If the solution to \(y^{\prime}=x+4 y\) passes through the point \((3,1),\) what is the slope of the solution at that point?

Solve and check each first-order linear differential equation. $$ y^{\prime}-y=2 $$

Verify that \(y(t)=M\left(1-e^{-a t}\right)\) solves the differential equation for limited growth, \(y^{\prime}=a(M-y),\) with initial condition \(y(0)=0\).

Verify that the function \(y\) satisfies the given differential equation. $$ \begin{array}{l} y=e^{5 x}-4 e^{x}+1 \\ y^{\prime \prime}-6 y^{\prime}+5 y=5 \end{array} $$

For the initial value problem \(\left\\{\begin{array}{l}y^{\prime}=4 x y \\\ y(1)=3\end{array}\right.\) state the initial point \(\left(x_{0}, y_{0}\right)\) and calculate the slope of the solution at this point.

Solve and check each first-order linear differential equation. $$ y^{\prime}-2 y=e^{-2 x} $$

Determine the type of each differential equation: unlimited growth, limited growth, logistic growth, or none of these. (Do not solve, just identify the type.) $$ y^{\prime}=0.02 y $$

Verify that the function \(y\) satisfies the given differential equation. $$ \begin{aligned} &y=k e^{a x}-\frac{b}{a} \quad(\text { for constants } a, b, \text { and } k)\\\ &y^{\prime}=a y+b \end{aligned} $$

For the initial value problem \(\left\\{\begin{array}{l}y^{\prime}=x / y \\\ y(6)=2\end{array}\right.\) state the initial point \(\left(x_{0}, y_{0}\right)\) and calculate the slope of the solution at this point.

Solve and check each first-order linear differential equation. $$ y^{\prime}+3 y=e^{3 x} $$

Determine the type of each differential equation: unlimited growth, limited growth, logistic growth, or none of these. (Do not solve, just identify the type.) $$ y^{\prime}=5(100-y) $$

Verify that the function \(y\) satisfies the given differential equation. $$ \begin{aligned} &y=a x^{2}+b x \quad \text { (for constants } a \text { and } \left.b\right)\\\ &y^{\prime}=\frac{y}{x}+a x \end{aligned} $$

For each initial value problem, calculate the Euler approximation for the solution on the interval [0,1] using \(n=4\) segments. Draw the graph of your approximation. (Carry out the calculations "by hand" with the aid of a calculator, rounding to two decimal places. Answers may differ slightly, depending on when you do the rounding.) $$ \begin{array}{l} y^{\prime}=3 x-2 y \\ y(0)=2 \end{array} $$

Solve each first-order linear differential equation. $$ y^{\prime}+\frac{5}{x} y=24 x^{2} $$

Determine the type of each differential equation: unlimited growth, limited growth, logistic growth, or none of these. (Do not solve, just identify the type.) $$ y^{\prime}=30(0.5-y) $$

Find the general solution of each differential equation or state that the differential equation is not separable. If the exercise says "and check," verify that your answer is a solution. $$ y^{2} y^{\prime}=4 x $$

For each initial value problem, calculate the Euler approximation for the solution on the interval [0,1] using \(n=4\) segments. Draw the graph of your approximation. (Carry out the calculations "by hand" with the aid of a calculator, rounding to two decimal places. Answers may differ slightly, depending on when you do the rounding.) $$ \begin{array}{l} y^{\prime}=x+2 y \\ y(0)=1 \end{array} $$

Solve each first-order linear differential equation. $$ y^{\prime}+\frac{4}{x} y=12 x $$

Determine the type of each differential equation: unlimited growth, limited growth, logistic growth, or none of these. (Do not solve, just identify the type.) $$ y^{\prime}=0.4 y(0.01-y) $$

Find the general solution of each differential equation or state that the differential equation is not separable. If the exercise says "and check," verify that your answer is a solution. $$ y^{4} y^{\prime}=8 x $$

For each initial value problem, calculate the Euler approximation for the solution on the interval [0,1] using \(n=4\) segments. Draw the graph of your approximation. (Carry out the calculations "by hand" with the aid of a calculator, rounding to two decimal places. Answers may differ slightly, depending on when you do the rounding.) $$ \begin{array}{l} y^{\prime}=4 x y \\ y(0)=1 \end{array} $$

Solve each first-order linear differential equation. $$ x y^{\prime}-y=x^{2} $$

Determine the type of each differential equation: unlimited growth, limited growth, logistic growth, or none of these. (Do not solve, just identify the type.) $$ y^{\prime}=2 y^{2}(0.5-y) $$

Find the general solution of each differential equation or state that the differential equation is not separable. If the exercise says "and check," verify that your answer is a solution. $$ y^{\prime}=x+y $$

For each initial value problem, calculate the Euler approximation for the solution on the interval [0,1] using \(n=4\) segments. Draw the graph of your approximation. (Carry out the calculations "by hand" with the aid of a calculator, rounding to two decimal places. Answers may differ slightly, depending on when you do the rounding.) $$ \begin{array}{l} y^{\prime}=8 x^{2}-y \\ y(0)=2 \end{array} $$

Solve each first-order linear differential equation. $$ x y^{\prime}-y=x $$

Determine the type of each differential equation: unlimited growth, limited growth, logistic growth, or none of these. (Do not solve, just identify the type.) $$ y^{\prime}=6 y $$

Find the general solution of each differential equation or state that the differential equation is not separable. If the exercise says "and check," verify that your answer is a solution. $$ y^{\prime}=x+y $$

For each initial value problem, calculate the Euler approximation for the solution on the interval [0,1] using \(n=4\) segments. Draw the graph of your approximation. (Carry out the calculations "by hand" with the aid of a calculator, rounding to two decimal places. Answers may differ slightly, depending on when you do the rounding.) $$ \begin{array}{l} y^{\prime}+2 y=e^{4 x} \\ y(0)=2 \end{array} $$

Solve each first-order linear differential equation. $$ y^{\prime}+3 x^{2} y=9 x^{2} $$

Determine the type of each differential equation: unlimited growth, limited growth, logistic growth, or none of these. (Do not solve, just identify the type.) $$ y^{\prime}=y(6-y) $$

Find the general solution of each differential equation or state that the differential equation is not separable. If the exercise says "and check," verify that your answer is a solution. $$ y^{\prime}=6 x^{2} y \quad \text { and check } $$

For each initial value problem, calculate the Euler approximation for the solution on the interval [0,1] using \(n=4\) segments. Draw the graph of your approximation. (Carry out the calculations "by hand" with the aid of a calculator, rounding to two decimal places. Answers may differ slightly, depending on when you do the rounding.) $$ \begin{array}{l} y^{\prime}+e^{y}=8 x \\ y(0)=0 \end{array} $$

Solve each first-order linear differential equation. $$ y^{\prime}-4 x^{3} y=8 x^{3} $$

Determine the type of each differential equation: unlimited growth, limited growth, logistic growth, or none of these. (Do not solve, just identify the type.) $$ y^{\prime}=0.01\left(100-y^{2}\right) $$

Find the general solution of each differential equation or state that the differential equation is not separable. If the exercise says "and check," verify that your answer is a solution. $$ y^{\prime}=12 x^{3} y \quad \text { and check } $$

For each initial value problem: a. Use an Euler's method graphing calculator program to find the estimate for \(y(2)\). Use the interval [0,2] with \(n=50\) segments. b. Solve the differential equation and initial condition exactly by separating variables or using an integrating factor. c. Evaluate the solution that you found in part (b) at \(x=2\). Compare this actual value of \(y(2)\) with the estimate of \(y(2)\) that you found in part (a). $$ \begin{array}{l} y^{\prime}=\frac{x}{y} \\ y(0)=1 \end{array} $$

Solve each first-order linear differential equation. $$ y^{\prime}-2 x y=0 $$

Determine the type of each differential equation: unlimited growth, limited growth, logistic growth, or none of these. (Do not solve, just identify the type.) $$ y^{\prime}=4 y(0.04-y) $$

Find the general solution of each differential equation or state that the differential equation is not separable. If the exercise says "and check," verify that your answer is a solution. $$ \begin{aligned} &y^{\prime}=\frac{y}{x}\\\ &\text { and check } \end{aligned} $$

For each initial value problem: a. Use an Euler's method graphing calculator program to find the estimate for \(y(2)\). Use the interval [0,2] with \(n=50\) segments. b. Solve the differential equation and initial condition exactly by separating variables or using an integrating factor. c. Evaluate the solution that you found in part (b) at \(x=2\). Compare this actual value of \(y(2)\) with the estimate of \(y(2)\) that you found in part (a). $$ \begin{array}{l} y^{\prime}=-x y \\ y(0)=1 \end{array} $$

Solve each first-order linear differential equation. $$ y^{\prime}+x y=0 $$

Determine the type of each differential equation: unlimited growth, limited growth, logistic growth, or none of these. (Do not solve, just identify the type.) $$ y^{\prime}=4500(1-y) $$

Find the general solution of each differential equation or state that the differential equation is not separable. If the exercise says "and check," verify that your answer is a solution. $$ y^{\prime}=\frac{y^{2}}{x^{2}} \quad \text { and check } $$

For each initial value problem: a. Use an Euler's method graphing calculator program to find the estimate for \(y(2)\). Use the interval [0,2] with \(n=50\) segments. b. Solve the differential equation and initial condition exactly by separating variables or using an integrating factor. c. Evaluate the solution that you found in part (b) at \(x=2\). Compare this actual value of \(y(2)\) with the estimate of \(y(2)\) that you found in part (a). $$ \begin{array}{l} \frac{d y}{d x}=0.2 y \\ y(0)=1 \end{array} $$

Solve each first-order linear differential equation. $$ (x+1) y^{\prime}+y=2 x $$