Chapter 4

Verify that \(y_{1}\) and \(y_{2}\) are solutions of the given differential equation but that \(y=c_{1} y_{1}+c_{2} y_{2}\) is, in general, not a solution. $$\left(y^{\prime \prime}\right)^{2}=y^{2} ; \quad y_{1}=e^{x}, y_{2}=\cos x$$

Solve the given system of differential equations by systematic elimination. $$\begin{aligned} &\frac{d x}{d t}=2 x-y\\\ &\frac{d y}{d t}=x \end{aligned}$$

Solve the given differential equation. $$x^{2} y^{\prime \prime}-2 y=0$$

W rite the given differential equation in the form \(L(y)=g(x),\) where \(L\) is a linear differential operator with constant coefficients. If possible, factor \(L\). $$9 y^{\prime \prime}-4 y=\sin x$$

Solve the given differential equation by undetermined coefficients.In Problems \(1-26\) solve the given differential equation by undetermined coefficients. $$y^{\prime \prime}+3 y^{\prime}+2 y=6$$

Find the general solution of the given second-order differential equation. $$4 y^{\prime \prime}+y^{\prime}=0$$

The indicated function \(y_{1}(x)\) is a solution of the given differential equation. Use reduction of order or formula (5), as instructed, to find a second solution \(y_{2}(x)\). $$y^{\prime \prime}-4 y^{\prime}+4 y=0 ; \quad y_{1}=e^{2 x}$$

The given family of functions is the general solution of the differential equation on the indicated interval. Find a member of the family that is a solution of the initial-value problem. $$\begin{aligned}&y=c_{1} e^{x}+c_{2} e^{-x},(-\infty, \infty);\\\&y^{\prime \prime}-y=0, \quad y(0)=0, \quad y^{\prime}(0)=1\end{aligned}$$

Verify that \(y_{1}\) and \(y_{2}\) are solutions of the given differential equation but that \(y=c_{1} y_{1}+c_{2} y_{2}\) is, in general, not a solution. $$y y^{\prime \prime}=\frac{1}{2}\left(y^{\prime}\right)^{2} ; \quad y_{1}=1, y_{2}=x^{2}$$

Solve the given system of differential equations by systematic elimination. $$\begin{aligned} &\frac{d x}{d t}=4 x+7 y\\\ &\frac{d y}{d t}=x-2 y \end{aligned}$$

Solve the given differential equation. $$4 x^{2} y^{\prime \prime}+y=0$$

W rite the given differential equation in the form \(L(y)=g(x),\) where \(L\) is a linear differential operator with constant coefficients. If possible, factor \(L\). $$y^{\prime \prime}-5 y=x^{2}-2 x$$

Solve each differential equation by variation of parameters. $$y^{\prime \prime}+y=\tan x$$

Solve the given differential equation by undetermined coefficients.In Problems \(1-26\) solve the given differential equation by undetermined coefficients. $$4 y^{\prime \prime}+9 y=15$$

Find the general solution of the given second-order differential equation. $$y^{\prime \prime}-36 y=0$$

The indicated function \(y_{1}(x)\) is a solution of the given differential equation. Use reduction of order or formula (5), as instructed, to find a second solution \(y_{2}(x)\). $$y^{\prime \prime}+2 y^{\prime}+y=0 ; \quad y_{1}=x e^{-x}$$

The given family of functions is the general solution of the differential equation on the indicated interval. Find a member of the family that is a solution of the initial-value problem. $$\begin{aligned}&y=c_{1} e^{4 x}+c_{2} e^{-x},(-\infty, \infty);\\\&y^{\prime \prime}-3 y^{\prime}-4 y=0, \quad y(0)=1, \quad y^{\prime}(0)=2\end{aligned}$$

The dependent variable \(y\) is missing in the given differential equation. Proceed as in Example 1 and solve the equation by using the substitution \(u=y^{\prime}\) $$y^{\prime \prime}+\left(y^{\prime}\right)^{2}+1=0$$

Solve the given system of differential equations by systematic elimination. $$\begin{aligned} &\frac{d x}{d t}=-y+t\\\ &\frac{d y}{d t}=x-t \end{aligned}$$

Solve the given differential equation. $$x y^{\prime \prime}+y^{\prime}=0$$

W rite the given differential equation in the form \(L(y)=g(x),\) where \(L\) is a linear differential operator with constant coefficients. If possible, factor \(L\). $$y^{\prime \prime}-4 y^{\prime}-12 y=x-6$$

Solve each differential equation by variation of parameters. $$y^{\prime \prime}+y=\sin x$$

Solve the given differential equation by undetermined coefficients.In Problems \(1-26\) solve the given differential equation by undetermined coefficients. $$y^{\prime \prime}-10 y^{\prime}+25 y=30 x+3$$

Find the general solution of the given second-order differential equation. $$y^{\prime \prime}-y^{\prime}-6 y=0$$

The indicated function \(y_{1}(x)\) is a solution of the given differential equation. Use reduction of order or formula (5), as instructed, to find a second solution \(y_{2}(x)\). $$y^{\prime \prime}+16 y=0 ; \quad y_{1}=\cos 4 x$$

The given family of functions is the general solution of the differential equation on the indicated interval. Find a member of the family that is a solution of the initial-value problem. $$\begin{aligned}&y=c_{1} x+c_{2} x \ln x,(0, \infty);\\\&x^{2} y^{\prime \prime}-x y^{\prime}+y=0, \quad y(1)=3, \quad y^{\prime}(1)=-1 \end{aligned}$$

The dependent variable \(y\) is missing in the given differential equation. Proceed as in Example 1 and solve the equation by using the substitution \(u=y^{\prime}\) $$y^{\prime \prime}=1+\left(y^{\prime}\right)^{2}$$

Solve the given system of differential equations by systematic elimination. $$\begin{aligned} &\frac{d x}{d t}-4 y=1\\\ &\frac{d y}{d t}+x=2 \end{aligned}$$

Solve the given differential equation. $$x y^{\prime \prime}-3 y^{\prime}=0$$

W rite the given differential equation in the form \(L(y)=g(x),\) where \(L\) is a linear differential operator with constant coefficients. If possible, factor \(L\). $$2 y^{\prime \prime}-3 y^{\prime}-2 y=1$$

Solve the given differential equation by undetermined coefficients.In Problems \(1-26\) solve the given differential equation by undetermined coefficients. $$y^{\prime \prime}+y^{\prime}-6 y=2 x$$

Find the general solution of the given second-order differential equation. $$y^{\prime \prime}-3 y^{\prime}+2 y=0$$

The indicated function \(y_{1}(x)\) is a solution of the given differential equation. Use reduction of order or formula (5), as instructed, to find a second solution \(y_{2}(x)\). $$y^{\prime \prime}+9 y=0 ; \quad y_{1}=\sin 3 x$$

The given family of functions is the general solution of the differential equation on the indicated interval. Find a member of the family that is a solution of the initial-value problem. $$\begin{aligned}&y=c_{1}+c_{2} \cos x+c_{3} \sin x,(-\infty, \infty);\\\&y^{\prime \prime \prime}+y^{\prime}=0, \quad y(\pi)=0, \quad y^{\prime}(\pi)=2, \quad y^{\prime \prime}(\pi)=-1\end{aligned}$$

The dependent variable \(y\) is missing in the given differential equation. Proceed as in Example 1 and solve the equation by using the substitution \(u=y^{\prime}\) $$x^{2} y^{\prime \prime}+\left(y^{\prime}\right)^{2}=0$$

Solve the given system of differential equations by systematic elimination. $$\left(D^{2}+5\right) x-\quad 2 y=0$$ $$-2 x+\left(D^{2}+2\right) y=0$$

Solve the given differential equation. $$x^{2} y^{\prime \prime}+x y^{\prime}+4 y=0$$

W rite the given differential equation in the form \(L(y)=g(x),\) where \(L\) is a linear differential operator with constant coefficients. If possible, factor \(L\). $$y^{\prime \prime \prime}+10 y^{\prime \prime}+25 y^{\prime}=e^{x}$$

Solve each differential equation by variation of parameters. $$y^{\prime \prime}+y=\cos ^{2} x$$

Solve the given differential equation by undetermined coefficients.In Problems \(1-26\) solve the given differential equation by undetermined coefficients. $$\frac{1}{4} y^{\prime \prime}+y^{\prime}+y=x^{2}-2 x$$

Find the general solution of the given second-order differential equation. $$y^{\prime \prime}+8 y^{\prime}+16 y=0$$

The indicated function \(y_{1}(x)\) is a solution of the given differential equation. Use reduction of order or formula (5), as instructed, to find a second solution \(y_{2}(x)\). $$y^{\prime \prime}-y=0 ; \quad y_{1}=\cosh x$$

Given that \(y=c_{1}+c_{2} x^{2}\) is a two-parameter family of solutions of \(x y^{\prime \prime}-y^{\prime}=0\) on the interval \((-\infty, \infty),\) show that constants \(c_{1}\) and \(c_{2}\) cannot be found so that a member of the family satisfies the initial conditions \(y(0)=0, y^{\prime}(0)=1 .\) Explain why this does not violate Theorem 4.1 .1.

The dependent variable \(y\) is missing in the given differential equation. Proceed as in Example 1 and solve the equation by using the substitution \(u=y^{\prime}\) $$e^{-x} y^{\prime \prime}=\left(y^{\prime}\right)^{2}$$

Solve the given system of differential equations by systematic elimination. $$\begin{aligned} (D+1) x+(D-1) y &=2 \\ 3 x+(D+2) y &=-1 \end{aligned}$$

Solve the given differential equation. $$x^{2} y^{\prime \prime}+5 x y^{\prime}+3 y=0$$

W rite the given differential equation in the form \(L(y)=g(x),\) where \(L\) is a linear differential operator with constant coefficients. If possible, factor \(L\). $$y^{\prime \prime \prime}+4 y^{\prime}=e^{x} \cos 2 x$$

Solve each differential equation by variation of parameters. $$y^{\prime \prime}+y=\sec ^{2} x$$

Solve the given differential equation by undetermined coefficients.In Problems \(1-26\) solve the given differential equation by undetermined coefficients. $$y^{\prime \prime}-8 y^{\prime}+20 y=100 x^{2}-26 x e^{x}$$

Find the general solution of the given second-order differential equation. $$y^{\prime \prime}-10 y^{\prime}+25 y=0$$