Chapter 7

Classify each of the following equations as linear or nonlinear. If the equation is linear, determine whether it is homogeneous or nonhomogeneous. $$ x^{3} y^{\prime \prime}+(x-1) y^{\prime}-8 y=0 $$

Classify each of the following equations as linear or nonlinear. If the equation is linear, determine whether it is homogeneous or nonhomogeneous. $$ \left(1+y^{2}\right) y^{\prime \prime}+x y^{\prime}-3 y=\cos x $$

Classify each of the following equations as linear or nonlinear. If the equation is linear, determine whether it is homogeneous or nonhomogeneous. $$ x y^{\prime \prime}+e^{y} y^{\prime}=x $$

Classify each of the following equations as linear or nonlinear. If the equation is linear, determine whether it is homogeneous or nonhomogeneous. $$ y^{\prime \prime}+\frac{4}{x} y^{\prime}-8 x y=5 x^{2}+1 $$

Classify each of the following equations as linear or nonlinear. If the equation is linear, determine whether it is homogeneous or nonhomogeneous. $$ y^{\prime \prime}+(\sin x) y^{\prime}-x y=4 y $$

Classify each of the following equations as linear or nonlinear. If the equation is linear, determine whether it is homogeneous or nonhomogeneous. $$ y^{\prime \prime}+\left(\frac{x+3}{y}\right) y^{\prime}=0 $$

For each of the following problems, verify that the given function is a solution to the differential equation. Use a graphing utility to graph the particular solutions for several values of \(c_{1}\) and \(c_{2} .\) What do the solutions have in common? $$ [\mathrm{T}] \mathrm{y}^{\prime \prime}+2 y^{\prime}-3 y=0 ; \quad y(x)=c_{1} e^{x}+c_{2} e^{-3 x} $$

For each of the following problems, verify that the given function is a solution to the differential equation. Use a graphing utility to graph the particular solutions for several values of \(c_{1}\) and \(c_{2} .\) What do the solutions have in common? $$ [\mathrm{T}] x^{2} y^{\prime \prime}-2 y-3 x^{2}+1=0 $$ $$ y(x)=c_{1} x^{2}+c_{2} x^{-1}+x^{2} \ln (x)+\frac{1}{2} $$

For each of the following problems, verify that the given function is a solution to the differential equation. Use a graphing utility to graph the particular solutions for several values of \(c_{1}\) and \(c_{2} .\) What do the solutions have in common? $$ [\mathbf{T}] y^{\prime \prime}+14 y^{\prime}+49 y=0 ; \quad y(x)=c_{1} e^{-7 x}+c_{2} x e^{-7 x} $$

For each of the following problems, verify that the given function is a solution to the differential equation. Use a graphing utility to graph the particular solutions for several values of \(c_{1}\) and \(c_{2} .\) What do the solutions have in common? $$ [\mathrm{T}] 6 y^{\prime \prime}-49 y^{\prime}+8 y=0 ; \quad y(x)=c_{1} e^{x / 6}+c_{2} e^{8 x} $$

Find the general solution to the linear differential equation. $$ y^{\prime \prime}-3 y^{\prime}-10 y=0 $$

Find the general solution to the linear differential equation. $$ y^{\prime \prime}-7 y^{\prime}+12 y=0 $$

Find the general solution to the linear differential equation. $$ y^{\prime \prime}+4 y^{\prime}+4 y=0 $$

Find the general solution to the linear differential equation. $$ 4 y^{\prime \prime}-12 y^{\prime}+9 y=0 $$

Find the general solution to the linear differential equation. $$ 2 y^{\prime \prime}-3 y^{\prime}-5 y=0 $$

Find the general solution to the linear differential equation. $$ 3 y^{\prime \prime}-14 y^{\prime}+8 y=0 $$

Find the general solution to the linear differential equation. $$ y^{\prime \prime}+y^{\prime}+y=0 $$

Find the general solution to the linear differential equation. $$ 5 y^{\prime \prime}+2 y^{\prime}+4 y=0 $$

Find the general solution to the linear differential equation. $$ y^{\prime \prime}-121 y=0 $$

Find the general solution to the linear differential equation. $$ 8 y^{\prime \prime}+14 y^{\prime}-15 y=0 $$

Find the general solution to the linear differential equation. $$ y^{\prime \prime}+81 y=0 $$

Find the general solution to the linear differential equation. $$ y^{\prime \prime}-y^{\prime}+11 y=0 $$

Find the general solution to the linear differential equation. $$ 2 y^{\prime \prime}=0 $$

Find the general solution to the linear differential equation. $$ y^{\prime \prime}-6 y^{\prime}+9 y=0 $$

Find the general solution to the linear differential equation. $$ 3 y^{\prime \prime}-2 y^{\prime}-7 y=0 $$

Find the general solution to the linear differential equation. $$ 4 y^{\prime \prime}-10 y^{\prime}=0 $$

Find the general solution to the linear differential equation. $$ 36 \frac{d^{2} y}{d x^{2}}+12 \frac{d y}{d x}+y=0 $$

Find the general solution to the linear differential equation. $$ 25 \frac{d^{2} y}{d x^{2}}-80 \frac{d y}{d x}+64 y=0 $$

Find the general solution to the linear differential equation. $$ \frac{d^{2} y}{d x^{2}}-9 \frac{d y}{d x}=0 $$

Find the general solution to the linear differential equation. $$ 4 \frac{d^{2} y}{d x^{2}}+8 y=0 $$

Solve the initial-value problem. $$ y^{\prime \prime}+5 y^{\prime}+6 y=0, \quad y(0)=0, \quad y^{\prime}(0)=-2 $$

Solve the initial-value problem. $$ y^{\prime \prime}+2 y^{\prime}-8 y=0, \quad y(0)=5, \quad y^{\prime}(0)=4 $$

Solve the initial-value problem. $$ y^{\prime \prime}+4 y=0, \qquad y(0)=3, \quad y^{\prime}(0)=10 $$

Solve the initial-value problem. $$ y^{\prime \prime}-18 y^{\prime}+81 y=0, \quad y(0)=1, \quad y^{\prime}(0)=5 $$

Solve the initial-value problem. $$ y^{\prime \prime}-y^{\prime}-30 y=0, \quad y(0)=1, \quad y^{\prime}(0)=-16 $$

Solve the initial-value problem. $$ 4 y^{\prime \prime}+4 y^{\prime}-8 y=0, \quad y(0)=2, \quad y^{\prime}(0)=1 $$

Solve the initial-value problem. $$ 25 y^{\prime \prime}+10 y^{\prime}+y=0, \quad y(0)=2, \quad y^{\prime}(0)=1 $$

Solve the initial-value problem. $$ y^{\prime \prime}+y=0, \quad y(\pi)=1, \quad y^{\prime}(\pi)=-5 $$

Solve the boundary-value problem, if possible. $$ y^{\prime \prime}+y^{\prime}-42 y=0, \quad y(0)=0, \quad y(1)=2 $$

Solve the boundary-value problem, if possible. $$ 9 y^{\prime \prime}+y=0, \quad y\left(\frac{3 \pi}{2}\right)=6, y(0)=-8 $$

Solve the boundary-value problem, if possible. $$ y^{\prime \prime}+10 y^{\prime}+34 y=0, \quad y(0)=6, \quad y(\pi)=2 $$

Solve the boundary-value problem, if possible. $$ y^{\prime \prime}+7 y^{\prime}-60 y=0, \qquad y(0)=4, \quad y(2)=0 $$

Solve the boundary-value problem, if possible. $$ y^{\prime \prime}-4 y^{\prime}+4 y=0, \quad y(0)=2, \quad y(1)=-1 $$

Solve the boundary-value problem, if possible. $$ y^{\prime \prime}-5 y^{\prime}=0, \qquad y(0)=3, \quad y(-1)=2 $$

Solve the boundary-value problem, if possible. $$ y^{\prime \prime}+9 y=0, \quad y(0)=4, \quad y\left(\frac{\pi}{3}\right)=-4 $$

Solve the boundary-value problem, if possible. $$ 4 y^{\prime \prime}+25 y=0, \quad y(0)=2, \quad y(2 \pi)=-2 $$

Find a differential equation with a general solution that is \(y=c_{1} e^{x / 5}+c_{2} e^{-4 x}\)

Find a differential equation with a general solution that is \(y=c_{1} e^{x}+c_{2} e^{-4 x / 3}\)

For each of the following differential equations: a. Solve the initial value problem. b. [T] Use a graphing utility to graph the particular solution. $$ y^{\prime \prime}+64 y=0 ; \quad y(0)=3, \quad y^{\prime}(0)=16 $$

For each of the following differential equations: a. Solve the initial value problem. b. [T] Use a graphing utility to graph the particular solution. $$ y^{\prime \prime}-2 y^{\prime}+10 y=0 \qquad y(0)=1, \quad y^{\prime}(0)=13 $$