Chapter 4

Find linearly independent functions that are annihilated by the given differential operator. $$(D-6)(2 D+3)$$

Solve the given initial-value problem. $$y^{\prime \prime}+16 y=0, \quad y(0)=2, y^{\prime}(0)=-2$$

Solve the given initial-value problem. $$y^{\prime \prime}+4 y^{\prime}+4 y=(3+x) e^{-2 x}, \quad y(0)=2, y^{\prime}(0)=5$$

Solve the given initial-value problem. Use a graphing utility to graph the solution curve. $$x^{2} y^{\prime \prime}-5 x y^{\prime}+8 y=8 x^{6}, \quad y(\frac{1}{2})=0, y^{\prime}(\frac{1}{2})=0$$

Find linearly independent functions that are annihilated by the given differential operator. $$D^{2}-9 D-36$$

Solve the given initial-value problem. $$\frac{d^{2} y}{d \theta^{2}}+y=0, \quad y(\pi / 3)=0, y^{\prime}(\pi / 3)=2$$

Solve the given initial-value problem. $$y^{\prime \prime}+4 y^{\prime}+5 y=35 e^{-4 x}, \quad y(0)=-3, y^{\prime}(0)=1$$

Use the substitution \(x=e^{t}\) to transform the given Cauchy-Euler equation to a differential equation with constant coefficients. Solve the original equation by solving the new equation using the procedures in Sections \(4.3-4.5\) $$x^{2} y^{\prime \prime}+9 x y^{\prime}-20 y=0$$

Solve the given initial-value problem. $$\frac{d^{2} y}{d t^{2}}-4 \frac{d y}{d t}-5 y=0, \quad y(1)=0, y^{\prime}(1)=2$$

Verify that the given two-parameter family of functions is the general solution of the nonhomogeneous differential equation on the indicated interval. $$\begin{aligned}&y^{\prime \prime}-7 y^{\prime}+10 y=24 e^{x}\\\&y=c_{1} e^{2 x}+c_{2} e^{5 x}+6 e^{x},(-\infty, \infty)\end{aligned}$$

$$y^{\Solve the given initial-value problem. $$y^{\prime \prime}-y=\cosh x, \quad y(0)=2, y^{\prime}(0)=12$$

Use the substitution \(x=e^{t}\) to transform the given Cauchy-Euler equation to a differential equation with constant coefficients. Solve the original equation by solving the new equation using the procedures in Sections \(4.3-4.5\) $$x^{2} y^{\prime \prime}-9 x y^{\prime}+25 y=0$$

Find linearly independent functions that are annihilated by the given differential operator. $$D^{2}-6 D+10$$

Solve the given initial-value problem. $$4 y^{\prime \prime}-4 y^{\prime}-3 y=0, \quad y(0)=1, y^{\prime}(0)=5$$

Verify that the given two-parameter family of functions is the general solution of the nonhomogeneous differential equation on the indicated interval. $$\begin{aligned}&y^{\prime \prime}+y=\sec x\\\&y=c_{1} \cos x+c_{2} \sin x+x \sin x+(\cos x) \ln (\cos x),(-\pi / 2, \pi / 2)\end{aligned}$$

Solve the given initial-value problem. $$y^{\prime \prime}-y=\cosh x, \quad y(0)=2, y^{\prime}(0)=12$$

Use the substitution \(x=e^{t}\) to transform the given Cauchy-Euler equation to a differential equation with constant coefficients. Solve the original equation by solving the new equation using the procedures in Sections \(4.3-4.5\) $$x^{2} y^{\prime \prime \prime}+10 x y^{\prime}+8 y=x^{2}$$

Find linearly independent functions that are annihilated by the given differential operator. $$D^{3}-10 D^{2}+25 D$$

Verify that the given two-parameter family of functions is the general solution of the nonhomogeneous differential equation on the indicated interval. $$\begin{aligned}&y^{\prime \prime}-4 y^{\prime}+4 y=2 e^{2 x}+4 x-12\\\&y=c_{1} e^{2 x}+c_{2} x e^{2 x}+x^{2} e^{2 x}+x-2,(-\infty, \infty) \end{aligned}$$

Solve the given initial-value problem. $$\frac{d^{2} x}{d t^{2}}+\omega^{2} x=F_{0} \cos \gamma t, \quad x(0)=0, x^{\prime}(0)=0$$

Discuss how the methods of undetermined coefficients and variation of parameters can be combined to solve the given differential equation. Carry out your ideas. $$y^{\prime \prime}-2 y^{\prime}+y=4 x^{2}-3+x^{-1} e^{x}$$

Use the substitution \(x=e^{t}\) to transform the given Cauchy-Euler equation to a differential equation with constant coefficients. Solve the original equation by solving the new equation using the procedures in Sections \(4.3-4.5\) $$x^{2} y^{\prime \prime}-4 x y^{\prime}+6 y=\ln x^{2}$$

Find linearly independent functions that are annihilated by the given differential operator. $$D^{2}(D-5)(D-7)$$

Solve the given initial-value problem. $$y^{\prime \prime}-2 y^{\prime}+y=0, \quad y(0)=5, y^{\prime}(0)=10$$

Verify that the given two-parameter family of functions is the general solution of the nonhomogeneous differential equation on the indicated interval. $$\begin{aligned}&2 x^{2} y^{\prime \prime}+5 x y^{\prime}+y=x^{2}-x\\\&y=c_{1} x^{-1 / 2}+c_{2} x^{-1}+\frac{1}{15} x^{2}-\frac{1}{6} x,(0, \infty)\end{aligned}$$

Solve the given initial-value problem. $$\begin{aligned}&y^{\prime \prime \prime}-2 y^{\prime \prime}+y^{\prime}=2-24 e^{x}+40 e^{5 x},\quad y(0)=\frac{1}{2}, y^{\prime}(0)=\frac{5}{2}\\\&y^{\prime \prime}(0)=-\frac{9}{2}\end{aligned}$$

Solve the given differential equation by undetermined coefficients. $$y^{\prime \prime}-9 y=54$$

Solve the given initial-value problem. $$y^{\prime \prime \prime}+12 y^{\prime \prime}+36 y^{\prime}=0, \quad y(0)=0, y^{\prime}(0)=1, y^{\prime \prime}(0)=-7$$

(a) Verify that \(y_{p_{1}}=3 e^{2 x}\) and \(y_{p_{2}}=x^{2}+3 x\) are, respectively, particular solutions of \(y^{\prime \prime}-6 y^{\prime}+5 y=-9 e^{2 x}\) and \(y^{\prime \prime}-6 y^{\prime}+5 y=5 x^{2}+3 x-16\). (b) Use part (a) to find particular solutions of \(y^{\prime \prime}-6 y^{\prime}+5 y=5 x^{2}+3 x-16-9 e^{2 x}\) and \(\quad y^{\prime \prime}-6 y^{\prime}+5 y=-10 x^{2}-6 x+32+e^{2 x}\).

Solve the given initial-value problem. $$\begin{aligned}&y^{\prime \prime \prime}+8 y=2 x-5+8 e^{-2 x}, \quad y(0)=-5, y^{\prime}(0)=3\\\&y^{\prime \prime}(0)=-4\end{aligned}$$

Find the general solution of \(x^{4} y^{\prime \prime}+x^{3} y^{\prime}-4 x^{2} y=1\) given that \(y_{1}=x^{2}\) is a solution of the associated homogeneous equation.

Use the substitution \(x=e^{t}\) to transform the given Cauchy-Euler equation to a differential equation with constant coefficients. Solve the original equation by solving the new equation using the procedures in Sections \(4.3-4.5\) $$x^{3} y^{m}-3 x^{2} y^{\prime \prime}+6 x y^{\prime}-6 y=3+\ln x^{3}$$

Solve the given differential equation by undetermined coefficients. $$2 y^{\prime \prime}-7 y^{\prime}+5 y=-29$$

Solve the given initial-value problem. $$y^{\prime \prime \prime}+2 y^{\prime \prime}-5 y^{\prime}-6 y=0, \quad y(0)=y^{\prime}(0)=0, y^{\prime \prime}(0)=1$$

(a) \(\mathrm{By}\) inspection find a particular solution of\\[y^{\prime \prime}+2 y=10\\] (b) By inspection find a particular solution of \(y^{\prime \prime}+2 y=-4 x\). (c) Find a particular solution of \(y^{\prime \prime}+2 y=-4 x+10\). (d) Find a particular solution of \(y^{\prime \prime}+2 y=8 x+5\).

Solve the given boundary-value problem. $$y^{\prime \prime}+y=x^{2}+1, \quad y(0)=5, y(1)=0$$

Use the substitution \(t=-x\) to solve the given initial-value problem on the interval \((-\infty, 0).\) $$4 x^{2} y^{\prime \prime}+y=0, \quad y(-1)=2, y^{\prime}(-1)=4$$

Solve the given differential equation by undetermined coefficients. $$y^{\prime \prime}+y^{\prime}=3$$

Solve the given boundary-value problem. $$y^{\prime \prime}-2 y^{\prime}+2 y=2 x-2, \quad y(0)=0, y(\pi)=\pi$$

Use the substitution \(t=-x\) to solve the given initial-value problem on the interval \((-\infty, 0).\) $$x^{2} y^{\prime \prime}-4 x y^{\prime}+6 y=0, \quad y(-2)=8, y^{\prime}(-2)=0$$

Suppose that \(y_{1}=e^{x}\) and \(y_{2}=e^{-x}\) are two solutions of a homogeneous linear differential equation. Explain why \(y_{3}=\cosh x\) and \(y_{4}=\sinh x\) are also solutions of the equation.

Solve the given boundary-value problem. $$y^{\prime \prime}+3 y=6 x, \quad y(0)=0, y(1)+y^{\prime}(1)=0$$

Use \(y=(x-x_{0})^{m}\) to solve the given differential equation. $$(x+3)^{2} y^{\prime \prime}-8(x+3) y^{\prime}+14 y=0$$

Solve the given differential equation by undetermined coefficients. $$y^{\prime \prime}+4 y^{\prime}+4 y=2 x+6$$

(a) Verify that \(y_{1}=x^{3}\) and \(y_{2}=|x|^{3}\) are linearly independent solutions of the differential equation \(x^{2} y^{\prime \prime}-4 x y^{\prime}+6 y=0\) on the interval \((-\infty, \infty)\). (b) For the functions \(y_{1}\) and \(y_{2}\) in part (a), show that \(W\left(y_{1}, y_{2}\right)=0\) for every real number \(x .\) Does this result violate Theorem 4.1.3? Explain. (c) Verify that \(Y_{1}=x^{3}\) and \(Y_{2}=x^{2}\) are also linearly independent solutions of the differential equation in part (a) on the interval \((-\infty, \infty)\). (d) Besides the functions \(y_{1}, y_{2}, Y_{1},\) and \(Y_{2}\) in parts (a) and (c), find a solution of the differential equation that satisfies \\[y(0)=0, y^{\prime}(0)=0\\] (e) By the superposition principle, Theorem \(4.1 .2,\) both linear combinations \(y=c_{1} y_{1}+c_{2} y_{2}\) and \(Y=c_{1} Y_{1}+c_{2} Y_{2}\) are solutions of the differential equation. Discuss whether one, both, or neither of the linear combinations is a general solution of the differential equation on the interval \((-\infty, \infty)\).

Solve the given boundary-value problem. $$y^{\prime \prime}+3 y=6 x, \quad y(0)+y^{\prime}(0)=0, y(1)=0$$

Solve the given differential equation by undetermined coefficients. $$y^{\prime \prime}+3 y^{\prime}=4 x-5$$

Is the set of functions \(f_{1}(x)=e^{x+2}, f_{2}(x)=e^{x-3}\) linearly dependent or linearly independent on \((-\infty, \infty) ?\) Discuss.

Solve the given initial-value problem in which the input function \(g(x)\) is discontinuous. [Hint: Solve each problem on two intervals, and then find a solution so that \(y\) and \(y^{\prime}\) are continuous at \(x=\pi / 2\) (Problem 41 ) and at \(x=\pi\) (Problem 42 ). \(y^{\prime \prime}+4 y=g(x), \quad y(0)=1, y^{\prime}(0)=2, \quad\) where\\[g(x)=\left\\{\begin{array}{ll}\sin x, & 0 \leq x \leq \pi / 2 \\\0, & x>\pi / 2\end{array}\right.\\]

Solve the given differential equation by undetermined coefficients. $$y^{\prime \prime \prime \prime}+y^{\prime \prime}=8 x^{2}$$