Chapter 8

Solve the differential equation by separation of variables. Where reasonable, express the family of solutions as explicit functions of x. $$ \frac{d y}{d x}=\frac{y}{x} $$

Solve the differential equation by the method of integrating factors. $$ \frac{d y}{d x}+4 y=e^{-3 x} $$

Confirm that \(y=3 e^{x^{3}}\) is a solution of the initial-value problem \(y^{\prime}=3 x^{2} y, y(0)=3\)

Sketch the slope field for \(y^{\prime}=x y / 4\) at the 25 gridpoints \((x, y),\) where \(x=-2,-1, \ldots, 2\) and \(y=-2,-1, \ldots, 2\)

Solve the differential equation by separation of variables. Where reasonable, express the family of solutions as explicit functions of x. $$ \frac{d y}{d x}=2\left(1+y^{2}\right) x $$

Solve the differential equation by the method of integrating factors. $$ \frac{d y}{d x}+2 x y=x $$

Confirm that \(y=\frac{1}{4} x^{4}+2 \cos x+1\) is a solution of the initial- value problem \(y^{\prime}=x^{3}-2 \sin x, y(0)=3\)

Sketch the slope field for \(y^{\prime}+y=2\) at the 25 gridpoints \((x, y),\) where \(x=0,1, \ldots, 4\) and \(y=0,1, \ldots, 4\)

Solve the differential equation by separation of variables. Where reasonable, express the family of solutions as explicit functions of x. $$ \frac{\sqrt{1+x^{2}}}{1+y} \frac{d y}{d x}=-x $$

Solve the differential equation by the method of integrating factors. $$ y^{\prime}+y=\cos \left(e^{x}\right) $$

State the order of the differential equation, and confirm that the functions in the given family are solutions. $$ \begin{array}{l}{\text { (a) }(1+x) \frac{d y}{d x}=y ; y=c(1+x)} \\ {\text { (b) } y^{\prime \prime}+y=0 ; y=c_{1} \sin t+c_{2} \cos t}\end{array} $$

Solve the differential equation by separation of variables. Where reasonable, express the family of solutions as explicit functions of x. $$ \left(1+x^{4}\right) \frac{d y}{d x}=\frac{x^{3}}{y} $$

Solve the differential equation by the method of integrating factors. $$ 2 \frac{d y}{d x}+4 y=1 $$

State the order of the differential equation, and confirm that the functions in the given family are solutions. $$ \begin{array}{l}{\text { (a) } 2 \frac{d y}{d x}+y=x-1 ; \quad y=c e^{-x / 2}+x-3} \\ {\text { (b) } y^{\prime \prime}-y=0 ; \quad y=c_{1} e^{t}+c_{2} e^{-t}}\end{array} $$

Solve the differential equation by separation of variables. Where reasonable, express the family of solutions as explicit functions of x. $$ \left(2+2 y^{2}\right) y^{\prime}=e^{x} y $$

Solve the differential equation by the method of integrating factors. $$ \left(x^{2}+1\right) \frac{d y}{d x}+x y=0 $$

True-False Determine whether the statement is true or false. Explain your answer. The equation $$\left(\frac{d y}{d x}\right)^{2}=\frac{d y}{d x}+2 y$$ is an example of a second-order differential equation.

Solve the differential equation by separation of variables. Where reasonable, express the family of solutions as explicit functions of x. $$ y^{\prime}=-x y $$

Solve the differential equation by the method of integrating factors. $$ \frac{d y}{d x}+y+\frac{1}{1-e^{x}}=0 $$

True-False Determine whether the statement is true or false. Explain your answer. The differential equation $$\frac{d y}{d x}=2 y+1$$ has a solution that is constant.

Solve the differential equation by separation of variables. Where reasonable, express the family of solutions as explicit functions of x. $$ e^{-y} \sin x-y^{\prime} \cos ^{2} x=0 $$

Solve the initial-value problem. $$ x \frac{d y}{d x}+y=x, \quad y(1)=2 $$

True-False Determine whether the statement is true or false. Explain your answer. We expect the general solution of the differential equation $$\frac{d^{3} y}{d x^{3}}+3 \frac{d^{2} y}{d x^{2}}-\frac{d y}{d x}+4 y=0$$ to involve three arbitrary constants.

Use Euler's Method with the given step size \(\Delta x\) or \(\Delta t\) to approximate the solution of the initial-value problem over the stated interval. Present your answer as a table and as a graph. $$d y / d x=\sqrt[3]{y}, y(0)=1,0 \leq x \leq 4, \Delta x=0.5$$

Solve the differential equation by separation of variables. Where reasonable, express the family of solutions as explicit functions of x. $$ y^{\prime}-(1+x)\left(1+y^{2}\right)=0 $$

Solve the initial-value problem. $$ x \frac{d y}{d x}-y=x^{2}, \quad y(1)=-1 $$

True-False Determine whether the statement is true or false. Explain your answer. If every solution to a differential equation can be expressed in the form \(y=A e^{x+b}\) for some choice of constants \(A\) and \(b,\) then the differential equation must be of second order.

Use Euler's Method with the given step size \(\Delta x\) or \(\Delta t\) to approximate the solution of the initial-value problem over the stated interval. Present your answer as a table and as a graph. $$d y / d x=x-y^{2}, y(0)=1,0 \leq x \leq 2, \Delta x=0.25$$

Solve the differential equation by separation of variables. Where reasonable, express the family of solutions as explicit functions of x. $$ \frac{d y}{d x}-\frac{y^{2}-y}{\sin x}=0 $$

Solve the initial-value problem. $$ \frac{d y}{d x}-2 x y=2 x, \quad y(0)=3 $$

In each part, verify that the functions are solutions of the differential equation by substituting the functions into the equation. $$ \begin{array}{l}{y^{\prime \prime}+y^{\prime}-2 y=0} \\ {\text { (a) } e^{-2 x} \text { and } e^{x}} \\ {\text { (b) } c_{1} e^{-2 x}+c_{2} e^{x}\left(c_{1}, c_{2} \text { constants }\right)}\end{array} $$

Use Euler's Method with the given step size \(\Delta x\) or \(\Delta t\) to approximate the solution of the initial-value problem over the stated interval. Present your answer as a table and as a graph. $$d y / d t=\cos y, y(0)=1,0 \leq t \leq 2, \Delta t=0.5$$

Solve the differential equation by separation of variables. Where reasonable, express the family of solutions as explicit functions of x. $$ y-\frac{d y}{d x} \sec x=0 $$

Solve the initial-value problem. $$ \frac{d y}{d t}+y=2, \quad y(0)=1 $$

In each part, verify that the functions are solutions of the differential equation by substituting the functions into the equation. $$ \begin{array}{l}{y^{\prime \prime}-y^{\prime}-6 y=0} \\ {\text { (a) } e^{-2 x} \text { and } e^{3 x}} \\ {\text { (b) } c_{1} e^{-2 x}+c_{2} e^{3 x}\left(c_{1}, c_{2} \text { constants }\right)}\end{array} $$

Use Euler's Method with the given step size \(\Delta x\) or \(\Delta t\) to approximate the solution of the initial-value problem over the stated interval. Present your answer as a table and as a graph. $$d y / d t=e^{-y}, y(0)=0,0 \leq t \leq 1, \Delta t=0.1$$

Solve the initial-value problem by separation of variables. $$ y^{\prime}=\frac{3 x^{2}}{2 y+\cos y}, \quad y(0)=\pi $$

In each part, verify that the functions are solutions of the differential equation by substituting the functions into the equation. $$ \begin{array}{l}{y^{\prime \prime}-4 y^{\prime}+4 y=0} \\ {\text { (a) } e^{2 x} \text { and } x e^{2 x}} \\ {\text { (b) } c_{1} e^{2 x}+c_{2} x e^{2 x}\left(c_{1}, c_{2} \text { constants }\right)}\end{array} $$

Consider the initial-value problem $$ y^{\prime}=\sin \pi t, \quad y(0)=0 $$ Use Euler's Method with five steps to approximate \(y(1)\)

Solve the initial-value problem by separation of variables. $$ y^{\prime}-x e^{y}=2 e^{y}, \quad y(0)=0 $$

Determine whether the statement is true or false. Explain your answer. If the first-order linear differential equation $$ \frac{d y}{d x}+p(x) y=q(x) $$ has a solution that is a constant function, then \(q(x)\) is a constant multiple of \(p(x) .\)

In each part, verify that the functions are solutions of the differential equation by substituting the functions into the equation. $$ \begin{array}{l}{y^{\prime \prime}-8 y^{\prime}+16 y=0} \\ {\text { (a) } e^{4 x} \text { and } x e^{4 x}} \\ {\text { (b) } c_{1} e^{4 x}+c_{2} x e^{4 x}\left(c_{1}, c_{2} \text { constants }\right)}\end{array} $$

Solve the initial-value problem by separation of variables. $$ \frac{d y}{d t}=\frac{2 t+1}{2 y-2}, \quad y(0)=-1 $$

Determine whether the statement is true or false. Explain your answer. In a mixing problem, we expect the concentration of the dissolved substance within the tank to approach a finite limit over time.

In each part, verify that the functions are solutions of the differential equation by substituting the functions into the equation. $$ \begin{array}{l}{y^{\prime \prime}+4 y=0} \\ {\text { (a) } \sin 2 x \text { and } \cos 2 x} \\ {\text { (b) } c_{1} \sin 2 x+c_{2} \cos 2 x \quad\left(c_{1}, c_{2} \text { constants }\right)}\end{array} $$

True–False Determine whether the statement is true or false. Explain your answer. Every integral curve for the slope field \(d y / d x=e^{x y}\) is the graph of an increasing function of \(x .\)

Solve the initial-value problem by separation of variables. $$ y^{\prime} \cosh ^{2} x-y \cosh 2 x=0, \quad y(0)=3 $$

In each part, verify that the functions are solutions of the differential equation by substituting the functions into the equation. $$ \begin{array}{l}{y^{\prime \prime}+4 y^{\prime}+13 y=0} \\ {\text { (a) } e^{-2 x} \sin 3 x \text { and } e^{-2 x} \cos 3 x} \\ {\text { (b) } e^{-2 x}\left(c_{1} \sin 3 x+c_{2} \cos 3 x\right) \quad\left(c_{1}, c_{2} \text { constants }\right)}\end{array} $$

True–False Determine whether the statement is true or false. Explain your answer. Every integral curve for the slope field \(d y / d x=e^{y}\) is concave up.

(a) Sketch some typical integral curves of the differential equation \(y^{\prime}=y / 2 x\). (b) Find an equation for the integral curve that passes through the point \((2,1)\).