Answer: In 3134, proceed as in Example 6 to find a solution of the initial-value problem

In Problems 1-6, proceed as in Example 1 to find a particular solution \(y_{p}(x)\) of the given differential equation in the integral form (9). \(y^{\prime \prime}-16 y=f(x)\)

In Problems 1-6, proceed as in Example 1 to find a particular solution \(y_{p}(x)\) of the given differential equation in the integral form (9). \(y^{\prime \prime}+3 y^{\prime}-10 y=f(x)\)

In Problems 1-6, proceed as in Example 1 to find a particular solution \(y_{p}(x)\) of the given differential equation in the integral form (9). \(y^{\prime \prime}+2 y^{\prime}+y=f(x)\)

In Problems 1-6, proceed as in Example 1 to find a particular solution \(y_{p}(x)\) of the given differential equation in the integral form (9). \(4 y^{\prime \prime}-4 y^{\prime}+y=f(x)\)

In Problems 1-6, proceed as in Example 1 to find a particular solution \(y_{p}(x)\) of the given differential equation in the integral form (9). \(y^{\prime \prime}+9 y=f(x)\)

In Problems 1-6, proceed as in Example 1 to find a particular solution \(y_{p}(x)\) of the given differential equation in the integral form (9). \(y^{\prime \prime}-2 y^{\prime}+2 y=f(x)\)

In Problems 7-12, proceed as in Example 2 to find the general solution of the given differential equation. Use the results obtained in Problems 1-6. Do not evaluate the integral that defines \(y_{p}(x)\). \(y^{\prime \prime}-16 y=x e^{-2 x}\)

In Problems 7-12, proceed as in Example 2 to find the general solution of the given differential equation. Use the results obtained in Problems 1-6. Do not evaluate the integral that defines \(y_{p}(x)\). \(y^{\prime \prime}+3 y^{\prime}-10 y=x^{2}\)

In Problems 7-12, proceed as in Example 2 to find the general solution of the given differential equation. Use the results obtained in Problems 1-6. Do not evaluate the integral that defines \(y_{p}(x)\). \(y^{\prime \prime}+2 y^{\prime}+y=e^{-x}\)

In Problems 7-12, proceed as in Example 2 to find the general solution of the given differential equation. Use the results obtained in Problems 1-6. Do not evaluate the integral that defines \(y_{p}(x)\). \(4 y^{\prime \prime}-4 y^{\prime}+y=\arctan x\)

In Problems 7-12, proceed as in Example 2 to find the general solution of the given differential equation. Use the results obtained in Problems 1-6. Do not evaluate the integral that defines \(y_{p}(x)\). \(y^{\prime \prime}+9 y=x+\sin x\)

In Problems 7-12, proceed as in Example 2 to find the general solution of the given differential equation. Use the results obtained in Problems 1-6. Do not evaluate the integral that defines \(y_{p}(x)\). \(y^{\prime \prime}-2 y^{\prime}+2 y=\cos ^{2} x\)

In Problems 13-18, proceed as in Example 3 to find the solution of the given initial-value problem. Evaluate the integral that defines \(y_{p}(x)\). \(y^{\prime \prime}-4 y=e^{2 x}\), y(0)=0, \(y^{\prime}(0)=0\)

In Problems 13-18, proceed as in Example 3 to find the solution of the given initial-value problem. Evaluate the integral that defines \(y_{p}(x)\). \(y^{\prime \prime}-y^{\prime}=1\), y(0)=0, \(y^{\prime}(0)=0\)

In Problems 13-18, proceed as in Example 3 to find the solution of the given initial-value problem. Evaluate the integral that defines \(y_{p}(x)\). \(y^{\prime \prime}-10 y^{\prime}+25 y=e^{5 x}\), y(0)=0, \(y^{\prime}(0)=0\)

In Problems 13-18, proceed as in Example 3 to find the solution of the given initial-value problem. Evaluate the integral that defines \(y_{p}(x)\). \(y^{\prime \prime}+6 y^{\prime}+9 y=x\), y(0)=0, \(y^{\prime}(0)=0\)

In Problems 13-18, proceed as in Example 3 to find the solution of the given initial-value problem. Evaluate the integral that defines \(y_{p}(x)\). \(y^{\prime \prime}+y=\csc x \cot x\), \(y(\pi / 2)=0\), \(y^{\prime}(\pi / 2)=0\)

In Problems 13-18, proceed as in Example 3 to find the solution of the given initial-value problem. Evaluate the integral that defines \(y_{p}(x)\). \(y^{\prime \prime}+y=\sec ^{2} x\), \(y(\pi)=0\), \(y^{\prime}(\pi)=0\)

In Problems 19-30, proceed as in Example 5 to find a solution of the given initial-value problem. \(y^{\prime \prime}-4 y=e^{2 x}\), y(0)=1, \(y^{\prime}(0)=-4\)

In Problems 19-30, proceed as in Example 5 to find a solution of the given initial-value problem. \(y^{\prime \prime}-y^{\prime}=1\), y(0)=10, \(y^{\prime}(0)=1\)

In Problems 19-30, proceed as in Example 5 to find a solution of the given initial-value problem. \(y^{\prime \prime}-10 y^{\prime}+25 y=e^{5 x}\), y(0)=-1, \(y^{\prime}(0)=1\)

In Problems 19-30, proceed as in Example 5 to find a solution of the given initial-value problem. \(y^{\prime \prime}+6 y^{\prime}+9 y=x\), y(0)=1, \(y^{\prime}(0)=-3\)

In Problems 19-30, proceed as in Example 5 to find a solution of the given initial-value problem. \(y^{\prime \prime}+y=\csc x \cot x\), \(y(\pi / 2)=-\pi / 2\), \(y^{\prime}(\pi / 2)=-1\)

In Problems 19-30, proceed as in Example 5 to find a solution of the given initial-value problem. \(y^{\prime \prime}+y=\sec ^{2} x\), \(y(\pi)=\frac{1}{2}\), \(y^{\prime}(\pi)=-1\)

In Problems 19-30, proceed as in Example 5 to find a solution of the given initial-value problem. \(y^{\prime \prime}+3 y^{\prime}+2 y=\sin e^{x}\), y(0)=-1, \(y^{\prime}(0)=0\)

In Problems 19-30, proceed as in Example 5 to find a solution of the given initial-value problem. \(y^{\prime \prime}+3 y^{\prime}+2 y=\frac{1}{1+e^{x}}\), y(0)=0, \(y^{\prime}(0)=1\)

In Problems 19-30, proceed as in Example 5 to find a solution of the given initial-value problem. \(x^{2} y^{\prime \prime}-2 x y^{\prime}+2 y=x\), y(1)=2, \(y^{\prime}(1)=-1\)

In Problems 19-30, proceed as in Example 5 to find a solution of the given initial-value problem. \(x^{2} y^{\prime \prime}-2 x y^{\prime}+2 y=x \ln x\), y(1)=1, \(y^{\prime}(1)=0\)

In Problems 19-30, proceed as in Example 5 to find a solution of the given initial-value problem. \(x^{2} y^{\prime \prime}-6 y=\ln x\), y(1)=1, \(y^{\prime}(1)=3\)

In Problems 19-30, proceed as in Example 5 to find a solution of the given initial-value problem. \(x^{2} y^{\prime \prime}-x y^{\prime}+y=x^{2}\), y(1)=4, \(y^{\prime}(1)=3\)

In Problems 31-34, proceed as in Example 6 to find a solution of the initial-value problem with the given piecewise-defined forcing function. \(y^{\prime \prime}-y=f(x), y(0)=8, \(y^{\prime}(0)=2\), where \(f(x)= \begin{cases}-1, & x<0 \\ 1, & x \geq 0\end{cases}\)

In Problems 31-34, proceed as in Example 6 to find a solution of the initial-value problem with the given piecewise-defined forcing function. \(y^{\prime \prime}-y=f(x), y(0)=3, y^{\prime}(0)=2\), where \(f(x)= \begin{cases}0, & x<0 \\ x, & x \geq 0\end{cases}\)

In Problems 31-34, proceed as in Example 6 to find a solution of the initial-value problem with the given piecewise-defined forcing function. \(y^{\prime \prime}+y=f(x), y(0)=1, y^{\prime}(0)=-1\), where \(f(x)= \begin{cases}0, & x<0 \\ 10, & 0 \leq x \leq 3 \pi \\ 0, & x>3 \pi\end{cases}\)

In Problems 31-34, proceed as in Example 6 to find a solution of the initial-value problem with the given piecewise-defined forcing function. \(y^{\prime \prime}+y=f(x)\), y(0)=0, \(y^{\prime}(0)=1\), where \(f(x)= \begin{cases}0, & x<0 \\ \cos x, & 0 \leq x \leq 4 \pi \\ 0, & x>4 \pi\end{cases}\)

In Problems 35 and 36, (a) use (25) and (26) to find a solution of the boundary-value problem. (b) Verify that function \(y_{p}(x)\) satisfies the differential equations and both boundary conditions. \(y^{\prime \prime}=f(x)\), y(0)=0, y(1)=0

In Problems 35 and 36, (a) use (25) and (26) to find a solution of the boundary-value problem. (b) Verify that function \(y_{p}(x)\) satisfies the differential equations and both boundary conditions. \(y^{\prime \prime}=f(x), y(0)=0, \(y(1)+y^{\prime}(1)=0\)

In Problem 35 find a solution of the BVP when f(x)=1.

In Problem 36 find a solution of the BVP when f(x)=x.

In Problems 39-44, proceed as in Examples 7 and 8 to find a solution of the given boundary-value problem. \(y^{\prime \prime}+y=1\), y(0)=0, y(1)=0

In Problems 39-44, proceed as in Examples 7 and 8 to find a solution of the given boundary-value problem. \(y^{\prime \prime}+9 y=1\), y(0)=0, \(y^{\prime}(\pi)=0\)

In Problems 39-44, proceed as in Examples 7 and 8 to find a solution of the given boundary-value problem. \(y^{\prime \prime}-2 y^{\prime}+2 y=e^{x}\), y(0)=0, \(y(\pi / 2)=0\)

In Problems 39-44, proceed as in Examples 7 and 8 to find a solution of the given boundary-value problem. \(y^{\prime \prime}-y^{\prime}=e^{2 x}\), y(0)=0, y(1)=0

In Problems 39-44, proceed as in Examples 7 and 8 to find a solution of the given boundary-value problem. \(x^{2} y^{\prime \prime}+x y^{\prime}=1\), \(y\left(e^{-1}\right)=0\), y(1)=0

In Problems 39-44, proceed as in Examples 7 and 8 to find a solution of the given boundary-value problem. \(x^{2} y^{\prime \prime}-4 x y^{\prime}+6 y=x^{4}\), \(y(1)-y^{\prime}(1)=0\), y(3)=0

Discussion Problems Suppose the solution of the boundary-value problem \(y^{\prime \prime}+P y^{\prime}+Q y=f(x)\), y(a)=0, y(b)=0, a<b, is given by \(y_{p}(x)=\int_{a}^{b} G(x, t) f(t) d t\) where \(y_{1}(x)\) and \(y_{2}(x)\) are solutions of the associated homogeneous differential equation chosen in the construction of G(x, t) so that \(y_{1}(a)=0\) and \(y_{2}(b)=0\). Prove that the solution of the boundary-value problem with nonhomogeneous DE and boundary conditions, \(y^{\prime \prime}+P y^{\prime}+Q y=f(x)\), y(a)=A, y(b)=B is given by \(y(x)=y_{p}(x)+\frac{B}{y_{1}(b)} y_{1}(x)+\frac{A}{y_{2}(a)} y_{2}(x)\) [Hint: In your proof, you will have to show that \(y_{1}(b) \neq 0\) and \(y_{2}(a) \neq 0\). Reread the assumptions following (23).]

Use the result in Problem 45 to solve \(y^{\prime \prime}+y=1\), y(0)=5, y(1)=-10