Chapter 4

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

The given two-parameter family is a solution of the indicated differential equation on the interval \((-\infty, \infty) .\) Determine whether a member of the family can be found that satisfies the boundary conditions. \(y=c_{1} e^{x} \cos x+c_{2} e^{x} \sin x ; \quad y^{\prime \prime}-2 y^{\prime}+2 y=0\) (a) \(y(0)=1, \quad y^{\prime}(\pi)=0\) (b) \(y(0)=1, \quad y(\pi)=-1\) (c) \(y(0)=1, \quad y(\pi / 2)=1\) (d) \(y(0)=0, \quad y(\pi)=0\).

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

Find two solutions of the initial-value problem \\[\left(y^{\prime \prime}\right)^{2}+\left(y^{\prime}\right)^{2}=1, \quad y\left(\frac{\pi}{2}\right)=\frac{1}{2}, \quad y^{\prime}\left(\frac{\pi}{2}\right)=\frac{\sqrt{3}}{2}\\] Use a numerical solver to graph the solution curves.

Solve the given system of differential equations by systematic elimination. $$\begin{aligned} &\frac{d x}{d t}+\frac{d y}{d t} \quad=e^{t}\\\ &-\frac{d^{2} x}{d t^{2}}+\frac{d x}{d t}+x+y=0 \end{aligned}$$

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

Solve the given differential equation. $$x^{2} y^{\prime \prime}-7 x y^{\prime}+41 y=0$$

Verify that the given differential operator annihilates the indicated functions. $$D^{2}+64 ; \quad y=2 \cos 8 x-5 \sin 8 x$$

Find the general solution of the given second-order differential equation. $$2 y^{\prime \prime}-3 y^{\prime}+4 y=0$$

The given two-parameter family is a solution of the indicated differential equation on the interval \((-\infty, \infty) .\) Determine whether a member of the family can be found that satisfies the boundary conditions. \(y=c_{1} x^{2}+c_{2} x^{4}+3 ; \quad x^{2} y^{\prime \prime}-5 x y^{\prime}+8 y=24\) (a) \(y(-1)=0, \quad y(1)=4\) (b) \(y(0)=1, \quad y(1)=2\) (c) \(y(0)=3, \quad y(1)=0\) (d) \(y(1)=3, \quad y(2)=15\)

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

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

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

Find a linear differential operator that annihilates the given function. $$1+6 x-2 x^{3}$$

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

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

Determine whether the given set of functions is linearly independent on the interval \((-\infty, \infty)\). $$f_{1}(x)=x, \quad f_{2}(x)=x^{2}, \quad f_{3}(x)=4 x-3 x^{2}$$

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

Solve the given differential equation by undetermined coefficients.In Problems \(1-26\) solve the given differential equation by undetermined coefficients. $$y^{\prime \prime}-5 y^{\prime}=2 x^{3}-4 x^{2}-x+6$$

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

Find a linear differential operator that annihilates the given function. $$x^{3}(1-5 x)$$

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

Determine whether the given set of functions is linearly independent on the interval \((-\infty, \infty)\). $$f_{1}(x)=0, \quad f_{2}(x)=x, \quad f_{3}(x)=e^{x}$$

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

Proceed as in Example 3 and obtain the first six nonzero terms of a Taylor series solution, centered at \(0,\) of the given initial-value problem. Use a numerical solver and a graphing utility to compare the solution curve with the graph of the Taylor polynomial. $$y^{\prime \prime}=x+y^{2}, \quad y(0)=1, y^{\prime}(0)=1$$

Solve the given system of differential equations by systematic elimination. $$\begin{aligned} &D x=y\\\ &\begin{array}{l} D y=z \\ D z=x \end{array} \end{aligned}$$

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

Solve the given differential equation. $$x y^{(4)}+6 y^{\prime \prime \prime}=0$$

Find a linear differential operator that annihilates the given function. $$1+7 e^{2 x}$$

Find the general solution of the given higher order differential equation. $$y^{\prime \prime \prime}-5 y^{\prime \prime}+3 y^{\prime}+9 y=0$$

Solve the given system of differential equations by systematic elimination. $$D x+\quad z=e^{t}$$ $$\begin{array}{r} (D-1) x+D y+D z=0 \\ x+2 y+D z=e^{t} \end{array}$$

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

Solve the given differential equation. $$x^{4} y^{(4)}+6 x^{3} y^{\prime \prime \prime}+9 x^{2} y^{\prime \prime}+3 x y^{\prime}+y=0$$

Find a linear differential operator that annihilates the given function. $$x+3 x e^{6 x}$$

Find the general solution of the given higher order differential equation. $$y^{\prime \prime \prime}+3 y^{\prime \prime}-4 y^{\prime}-12 y=0$$

Determine whether the given set of functions is linearly independent on the interval \((-\infty, \infty)\). $$f_{1}(x)=\cos 2 x, \quad f_{2}(x)=1, \quad f_{3}(x)=\cos ^{2} x$$

Solve the given system of differential equations by systematic elimination. $$\begin{aligned} &\frac{d x}{d t}=6 y\\\ &\frac{d y}{d t}=x+z\\\ &\frac{d z}{d t}=x+y \end{aligned}$$

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

Solve each differential equation by variation of parameters, subject to the initial conditions \(y(0)=1, y^{\prime}(0)=0\). $$4 y^{\prime \prime}-y=x e^{x / 2}$$

Find a linear differential operator that annihilates the given function. $$\cos 2 x$$

The indicated function \(y_{1}(x)\) is a solution of the associated homogeneous equation. Use the method of reduction of order to find a second solution \(y_{2}(x)\) of the homogeneous equation and a particular solution \(y_{p}(x)\) of the given non-homogeneous equation. $$y^{\prime \prime}-3 y^{\prime}+2 y=5 e^{3 x} ; \quad y_{1}=e^{x}$$

Find the general solution of the given higher order differential equation. $$\frac{d^{3} u}{d t^{3}}+\frac{d^{2} u}{d t^{2}}-2 u=0$$

Determine whether the given set of functions is linearly independent on the interval \((-\infty, \infty)\). $$f_{1}(x)=x, \quad f_{2}(x)=x-1, \quad f_{3}(x)=x+3$$

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

Solve the given differential equation by undetermined coefficients.In Problems \(1-26\) solve the given differential equation by undetermined coefficients. $$y^{\prime \prime}+2 y^{\prime}-24 y=16-(x+2) e^{4 x}$$

Solve each differential equation by variation of parameters, subject to the initial conditions \(y(0)=1, y^{\prime}(0)=0\). $$2 y^{\prime \prime}+y^{\prime}-y=x+1$$

Find a linear differential operator that annihilates the given function. $$1+\sin x$$

The indicated function \(y_{1}(x)\) is a solution of the associated homogeneous equation. Use the method of reduction of order to find a second solution \(y_{2}(x)\) of the homogeneous equation and a particular solution \(y_{p}(x)\) of the given non-homogeneous equation. $$y^{\prime \prime}-4 y^{\prime}+3 y=x ; \quad y_{1}=e^{x}$$

Find the general solution of the given higher order differential equation. $$\frac{d^{3} x}{d t^{3}}-\frac{d^{2} x}{d t^{2}}-4 x=0$$

Determine whether the given set of functions is linearly independent on the interval \((-\infty, \infty)\). $$f_{1}(x)=2+x, \quad f_{2}(x)=2+|x|$$