Chapter 4

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 \(y_{2}(x)\). $$y^{\prime \prime}-25 y=0 ; \quad y_{1}=e^{5 x}$$

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

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

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 \(L\). $$y^{\prime \prime \prime}+2 y^{\prime \prime}-13 y^{\prime}+10 y=x e^{-x}$$

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

Solve the given differential equation by undetermined coefficients.In Problems \(1-26\) solve the given differential equation by undetermined coefficients. $$y^{\prime \prime}+3 y=-48 x^{2} e^{3 x}$$

Find the general solution of the given second-order differential equation. $$12 y^{\prime \prime}-5 y^{\prime}-2 y=0$$

Given that \(x(t)=c_{1} \cos \omega t+c_{2} \sin \omega t\) is the general solution of \(x^{\prime \prime}+\omega^{2} x=0\) on the interval \((-\infty, \infty),\) show that a solution satisfying the initial conditions \(x(0)=x_{0}, x^{\prime}(0)=x_{1}\) is given by \(x(t)=x_{0} \cos \omega t+\frac{x_{1}}{\omega} \sin \omega t\).

Solve the given system of differential equations by systematic elimination. $$\begin{aligned} &\frac{d^{2} x}{d t^{2}}+\frac{d y}{d t}=-5 x\\\ &\frac{d x}{d t}+\frac{d y}{d t}=-x+4 y \end{aligned}$$

Solve the given differential equation. $$x^{2} y^{\prime \prime}+3 x y^{\prime}-4 y=0$$

Solve the given differential equation by undetermined coefficients.In Problems \(1-26\) solve the given differential equation by undetermined coefficients. $$4 y^{\prime \prime}-4 y^{\prime}-3 y=\cos 2 x$$

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 \(L\). $$y^{\prime \prime \prime}+4 y^{\prime \prime}+3 y^{\prime}=x^{2} \cos x-3 x$$

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

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

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 \(y_{2}(x)\). $$6 y^{\prime \prime}+y^{\prime}-y=0 ; \quad y_{1}=e^{x / 3}$$

Solve the given system of differential equations by systematic elimination. $$D x+\quad D^{2} y=e^{3 t}$$ $$(D+1) x+(D-1) y=4 e^{3 t}$$

Solve the given differential equation. $$25 x^{2} y^{\prime \prime}+25 x y^{\prime}+y=0$$

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

Solve each differential equation by variation of parameters. $$y^{\prime \prime}-9 y=\frac{9 x}{e^{3 x}}$$

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 \(L\). $$y^{(4)}+8 y^{\prime}=4$$

Find the general solution of the given second-order differential equation. $$y^{\prime \prime}+9 y=0$$

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 \(y_{2}(x)\). $$x^{2} y^{\prime \prime}-7 x y^{\prime}+16 y=0 ; \quad y_{1}=x^{4}$$

Find an interval centered about \(x=0\) for which the given initial-value problem has a unique solution. $$(x-2) y^{\prime \prime}+3 y=x, \quad y(0)=0, \quad y^{\prime}(0)=1$$

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

Solve the given system of differential equations by systematic elimination. $$D^{2} x-\quad D y=t$$ $$(D+3) x+(D+3) y=2$$

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

Solve the given differential equation by undetermined coefficients.In Problems \(1-26\) solve the given differential equation by undetermined coefficients. $$y^{\prime \prime}+2 y^{\prime}=2 x+5-e^{-2 x}$$

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 \(L\). $$y^{(4)}-8 y^{\prime \prime}+16 y=\left(x^{3}-2 x\right) e^{4 x}$$

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

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 \(y_{2}(x)\). $$x^{2} y^{\prime \prime}+2 x y^{\prime}-6 y=0 ; \quad y_{1}=x^{2}$$

Find an interval centered about \(x=0\) for which the given initial-value problem has a unique solution. $$y^{\prime \prime}+(\tan x) y=e^{x}, \quad y(0)=1, \quad y^{\prime}(0)=0$$

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

Solve the given system of differential equations by systematic elimination. $$\begin{array}{l} \left(D^{2}-1\right) x-y=0 \\ (D-1) x+D y=0 \end{array}$$

Solve the given differential equation. $$x^{2} y^{\prime \prime}+5 x y^{\prime}+4 y=0$$

Solve the given differential equation by undetermined coefficients.In Problems \(1-26\) solve the given differential equation by undetermined coefficients. $$y^{\prime \prime}-y^{\prime}+\frac{1}{4} y=3+e^{x / 2}$$

Verify that the given differential operator annihilates the indicated functions. $$D^{4} ; \quad y=10 x^{3}-2 x$$

Find the general solution of the given second-order differential equation. $$y^{\prime \prime}-4 y^{\prime}+5 y=0$$

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 \(y_{2}(x)\). $$x y^{\prime \prime}+y^{\prime}=0 ; \quad y_{1}=\ln x$$

Solve the given initial-value problem. $$y^{\prime \prime}+x\left(y^{\prime}\right)^{2}=0, y(1)=4, y^{\prime}(1)=2$$

Solve the given system of differential equations by systematic elimination. $$\left(2 D^{2}-D-1\right) x-(2 D+1) y=1$$ $$(D-1) x+\quad D y=-1$$

Solve the given differential equation. $$x^{2} y^{\prime \prime}+8 x y^{\prime}+6 y=0$$

Solve the given differential equation by undetermined coefficients.In Problems \(1-26\) solve the given differential equation by undetermined coefficients. $$y^{\prime \prime}-16 y=2 e^{4 x}$$

Verify that the given differential operator annihilates the indicated functions. $$2 D-1 ; \quad y=4 e^{x / 2}$$

Solve each differential equation by variation of parameters. $$y^{\prime \prime}-2 y^{\prime}+y=\frac{e^{x}}{1+x^{2}}$$

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

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 \(y_{2}(x)\). $$4 x^{2} y^{\prime \prime}+y=0 ; \quad y_{1}=x^{1 / 2} \ln x$$

Solve the given system of differential equations by systematic elimination. $$\begin{aligned} 2 \frac{d x}{d t}-5 x+\frac{d y}{d t} &=e^{t} \\ \frac{d x}{d t}-x+\frac{d y}{d t} &=5 e^{t} \end{aligned}$$

Solve the given differential equation by undetermined coefficients.In Problems \(1-26\) solve the given differential equation by undetermined coefficients. $$y^{\prime \prime}-16 y=2 e^{4 x}$$

Solve the given differential equation. $$3 x^{2} y^{\prime \prime}+6 x y^{\prime}+y=0$$

Verify that the given differential operator annihilates the indicated functions. $$(D-2)(D+5) ; \quad y=e^{2 x}+3 e^{-5 x}$$