Internal problem ID [11825]
Internal file name [OUTPUT/11835_Thursday_April_11_2024_08_52_43_PM_27972420/index.tex]
Book: Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi.
2004.
Section: Chapter 4, Section 4.3. The method of undetermined coefficients. Exercises page
151
Problem number: 52.
ODE order: 4.
ODE degree: 1.
The type(s) of ODE detected by this program : "higher_order_linear_constant_coefficients_ODE"
Maple gives the following as the ode type
[[_high_order, _linear, _nonhomogeneous]]
\[ \boxed {y^{\prime \prime \prime \prime }+16 y=x \,{\mathrm e}^{\sqrt {2}\, x} \sin \left (\sqrt {2}\, x \right )+{\mathrm e}^{-\sqrt {2}\, x} \cos \left (\sqrt {2}\, x \right )} \] This is higher order nonhomogeneous ODE. Let the solution be \[ y = y_h + y_p \] Where \(y_h\) is the solution to the homogeneous ODE And \(y_p\) is a particular solution to the nonhomogeneous ODE. \(y_h\) is the solution to \[ y^{\prime \prime \prime \prime }+16 y = 0 \] The characteristic equation is \[ \lambda ^{4}+16 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= \sqrt {2}+i \sqrt {2}\\ \lambda _2 &= -\sqrt {2}+i \sqrt {2}\\ \lambda _3 &= -\sqrt {2}-i \sqrt {2}\\ \lambda _4 &= \sqrt {2}-i \sqrt {2} \end {align*}
Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{\left (-\sqrt {2}-i \sqrt {2}\right ) x} c_{1} +{\mathrm e}^{\left (\sqrt {2}-i \sqrt {2}\right ) x} c_{2} +{\mathrm e}^{\left (-\sqrt {2}+i \sqrt {2}\right ) x} c_{3} +{\mathrm e}^{\left (\sqrt {2}+i \sqrt {2}\right ) x} c_{4} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= {\mathrm e}^{\left (-\sqrt {2}-i \sqrt {2}\right ) x} \\ y_2 &= {\mathrm e}^{\left (\sqrt {2}-i \sqrt {2}\right ) x} \\ y_3 &= {\mathrm e}^{\left (-\sqrt {2}+i \sqrt {2}\right ) x} \\ y_4 &= {\mathrm e}^{\left (\sqrt {2}+i \sqrt {2}\right ) x} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\prime \prime \prime \prime }+16 y = x \,{\mathrm e}^{\sqrt {2}\, x} \sin \left (\sqrt {2}\, x \right )+{\mathrm e}^{-\sqrt {2}\, x} \cos \left (\sqrt {2}\, x \right ) \] Let the particular solution be \[ y_p = U_1 y_1+U_2 y_2+U_3 y_3+U_4 y_4 \] Where \(y_i\) are the basis solutions found above for the homogeneous solution \(y_h\) and \(U_i(x)\) are functions to be determined as follows \[ U_i = (-1)^{n-i} \int { \frac {F(x) W_i(x) }{a W(x)} \, dx} \] Where \(W(x)\) is the Wronskian and \(W_i(x)\) is the Wronskian that results after deleting the last row and the \(i\)-th column of the determinant and \(n\) is the order of the ODE or equivalently, the number of basis solutions, and \(a\) is the coefficient of the leading derivative in the ODE, and \(F(x)\) is the RHS of the ODE. Therefore, the first step is to find the Wronskian \(W \left (x \right )\). This is given by \begin {equation*} W(x) = \begin {vmatrix} y_1&y_2&y_3&y_4\\ y_1'&y_2'&y_3'&y_4'\\ y_1''&y_2''&y_3''&y_4''\\ y_1'''&y_2'''&y_3'''&y_4'''\\ \end {vmatrix} \end {equation*} Substituting the fundamental set of solutions \(y_i\) found above in the Wronskian gives \begin {align*} W &= \left [\begin {array}{cccc} {\mathrm e}^{\left (-1-i\right ) \sqrt {2}\, x} & {\mathrm e}^{\left (1-i\right ) \sqrt {2}\, x} & {\mathrm e}^{\left (-1+i\right ) \sqrt {2}\, x} & {\mathrm e}^{\left (1+i\right ) \sqrt {2}\, x} \\ \left (-1-i\right ) \sqrt {2}\, {\mathrm e}^{\left (-1-i\right ) \sqrt {2}\, x} & \left (1-i\right ) \sqrt {2}\, {\mathrm e}^{\left (1-i\right ) \sqrt {2}\, x} & \left (-1+i\right ) \sqrt {2}\, {\mathrm e}^{\left (-1+i\right ) \sqrt {2}\, x} & \left (1+i\right ) \sqrt {2}\, {\mathrm e}^{\left (1+i\right ) \sqrt {2}\, x} \\ 4 i {\mathrm e}^{\left (-1-i\right ) \sqrt {2}\, x} & -4 i {\mathrm e}^{\left (1-i\right ) \sqrt {2}\, x} & -4 i {\mathrm e}^{\left (-1+i\right ) \sqrt {2}\, x} & 4 i {\mathrm e}^{\left (1+i\right ) \sqrt {2}\, x} \\ \left (4-4 i\right ) \sqrt {2}\, {\mathrm e}^{\left (-1-i\right ) \sqrt {2}\, x} & \left (-4-4 i\right ) \sqrt {2}\, {\mathrm e}^{\left (1-i\right ) \sqrt {2}\, x} & \left (4+4 i\right ) \sqrt {2}\, {\mathrm e}^{\left (-1+i\right ) \sqrt {2}\, x} & \left (-4+4 i\right ) \sqrt {2}\, {\mathrm e}^{\left (1+i\right ) \sqrt {2}\, x} \end {array}\right ] \\ |W| &= 1024 \,{\mathrm e}^{\left (-1-i\right ) \sqrt {2}\, x} {\mathrm e}^{\left (1-i\right ) \sqrt {2}\, x} {\mathrm e}^{\left (1+i\right ) \sqrt {2}\, x} {\mathrm e}^{\left (-1+i\right ) \sqrt {2}\, x} \end {align*}
The determinant simplifies to \begin {align*} |W| &= 1024 \end {align*}
Now we determine \(W_i\) for each \(U_i\). \begin {align*} W_1(x) &= \det \,\left [\begin {array}{ccc} {\mathrm e}^{\left (1-i\right ) \sqrt {2}\, x} & {\mathrm e}^{\left (-1+i\right ) \sqrt {2}\, x} & {\mathrm e}^{\left (1+i\right ) \sqrt {2}\, x} \\ \left (1-i\right ) \sqrt {2}\, {\mathrm e}^{\left (1-i\right ) \sqrt {2}\, x} & \left (-1+i\right ) \sqrt {2}\, {\mathrm e}^{\left (-1+i\right ) \sqrt {2}\, x} & \left (1+i\right ) \sqrt {2}\, {\mathrm e}^{\left (1+i\right ) \sqrt {2}\, x} \\ -4 i {\mathrm e}^{\left (1-i\right ) \sqrt {2}\, x} & -4 i {\mathrm e}^{\left (-1+i\right ) \sqrt {2}\, x} & 4 i {\mathrm e}^{\left (1+i\right ) \sqrt {2}\, x} \end {array}\right ] \\ &= \left (-16-16 i\right ) \sqrt {2}\, {\mathrm e}^{\left (1+i\right ) \sqrt {2}\, x} \end {align*}
\begin {align*} W_2(x) &= \det \,\left [\begin {array}{ccc} {\mathrm e}^{\left (-1-i\right ) \sqrt {2}\, x} & {\mathrm e}^{\left (-1+i\right ) \sqrt {2}\, x} & {\mathrm e}^{\left (1+i\right ) \sqrt {2}\, x} \\ \left (-1-i\right ) \sqrt {2}\, {\mathrm e}^{\left (-1-i\right ) \sqrt {2}\, x} & \left (-1+i\right ) \sqrt {2}\, {\mathrm e}^{\left (-1+i\right ) \sqrt {2}\, x} & \left (1+i\right ) \sqrt {2}\, {\mathrm e}^{\left (1+i\right ) \sqrt {2}\, x} \\ 4 i {\mathrm e}^{\left (-1-i\right ) \sqrt {2}\, x} & -4 i {\mathrm e}^{\left (-1+i\right ) \sqrt {2}\, x} & 4 i {\mathrm e}^{\left (1+i\right ) \sqrt {2}\, x} \end {array}\right ] \\ &= \left (-16+16 i\right ) \sqrt {2}\, {\mathrm e}^{\left (-1+i\right ) \sqrt {2}\, x} \end {align*}
\begin {align*} W_3(x) &= \det \,\left [\begin {array}{ccc} {\mathrm e}^{\left (-1-i\right ) \sqrt {2}\, x} & {\mathrm e}^{\left (1-i\right ) \sqrt {2}\, x} & {\mathrm e}^{\left (1+i\right ) \sqrt {2}\, x} \\ \left (-1-i\right ) \sqrt {2}\, {\mathrm e}^{\left (-1-i\right ) \sqrt {2}\, x} & \left (1-i\right ) \sqrt {2}\, {\mathrm e}^{\left (1-i\right ) \sqrt {2}\, x} & \left (1+i\right ) \sqrt {2}\, {\mathrm e}^{\left (1+i\right ) \sqrt {2}\, x} \\ 4 i {\mathrm e}^{\left (-1-i\right ) \sqrt {2}\, x} & -4 i {\mathrm e}^{\left (1-i\right ) \sqrt {2}\, x} & 4 i {\mathrm e}^{\left (1+i\right ) \sqrt {2}\, x} \end {array}\right ] \\ &= \left (-16+16 i\right ) \sqrt {2}\, {\mathrm e}^{\left (1-i\right ) \sqrt {2}\, x} \end {align*}
\begin {align*} W_4(x) &= \det \,\left [\begin {array}{ccc} {\mathrm e}^{\left (-1-i\right ) \sqrt {2}\, x} & {\mathrm e}^{\left (1-i\right ) \sqrt {2}\, x} & {\mathrm e}^{\left (-1+i\right ) \sqrt {2}\, x} \\ \left (-1-i\right ) \sqrt {2}\, {\mathrm e}^{\left (-1-i\right ) \sqrt {2}\, x} & \left (1-i\right ) \sqrt {2}\, {\mathrm e}^{\left (1-i\right ) \sqrt {2}\, x} & \left (-1+i\right ) \sqrt {2}\, {\mathrm e}^{\left (-1+i\right ) \sqrt {2}\, x} \\ 4 i {\mathrm e}^{\left (-1-i\right ) \sqrt {2}\, x} & -4 i {\mathrm e}^{\left (1-i\right ) \sqrt {2}\, x} & -4 i {\mathrm e}^{\left (-1+i\right ) \sqrt {2}\, x} \end {array}\right ] \\ &= \left (-16-16 i\right ) \sqrt {2}\, {\mathrm e}^{\left (-1-i\right ) \sqrt {2}\, x} \end {align*}
Now we are ready to evaluate each \(U_i(x)\). \begin {align*} U_1 &= (-1)^{4-1} \int { \frac {F(x) W_1(x) }{a W(x)} \, dx}\\ &= (-1)^{3} \int { \frac { \left (x \,{\mathrm e}^{\sqrt {2}\, x} \sin \left (\sqrt {2}\, x \right )+{\mathrm e}^{-\sqrt {2}\, x} \cos \left (\sqrt {2}\, x \right )\right ) \left (\left (-16-16 i\right ) \sqrt {2}\, {\mathrm e}^{\left (1+i\right ) \sqrt {2}\, x}\right )}{\left (1\right ) \left (1024\right )} \, dx} \\ &= - \int { \frac {\left (-16-16 i\right ) \left (x \,{\mathrm e}^{\sqrt {2}\, x} \sin \left (\sqrt {2}\, x \right )+{\mathrm e}^{-\sqrt {2}\, x} \cos \left (\sqrt {2}\, x \right )\right ) \sqrt {2}\, {\mathrm e}^{\left (1+i\right ) \sqrt {2}\, x}}{1024} \, dx}\\ &= - \int {\left (\left (-\frac {1}{64}-\frac {i}{64}\right ) \sqrt {2}\, {\mathrm e}^{i \sqrt {2}\, x} \left (x \,{\mathrm e}^{2 \sqrt {2}\, x} \sin \left (\sqrt {2}\, x \right )+\cos \left (\sqrt {2}\, x \right )\right )\right ) \, dx} \\ &= \frac {\sqrt {2}\, x}{128}+\frac {i \sqrt {2}\, x}{128}+\frac {{\mathrm e}^{2 i \sqrt {2}\, x}}{256}-\frac {i {\mathrm e}^{2 i \sqrt {2}\, x}}{256}+\left (-\frac {1}{8192}-\frac {i}{8192}\right ) \sqrt {2}\, \left (\sqrt {2}+i \sqrt {2}\right ) \left (i \sqrt {2}\, {\mathrm e}^{2 i \sqrt {2}\, x}-{\mathrm e}^{2 i \sqrt {2}\, x} \sqrt {2}+8 x \,{\mathrm e}^{2 i \sqrt {2}\, x}+2 i \sqrt {2}-8 i x +2 \sqrt {2}-8 x \right ) {\mathrm e}^{2 \sqrt {2}\, x} \\ &= \frac {\sqrt {2}\, x}{128}+\frac {i \sqrt {2}\, x}{128}+\frac {{\mathrm e}^{2 i \sqrt {2}\, x}}{256}-\frac {i {\mathrm e}^{2 i \sqrt {2}\, x}}{256}+\left (-\frac {1}{8192}-\frac {i}{8192}\right ) \sqrt {2}\, \left (\sqrt {2}+i \sqrt {2}\right ) \left (i \sqrt {2}\, {\mathrm e}^{2 i \sqrt {2}\, x}-{\mathrm e}^{2 i \sqrt {2}\, x} \sqrt {2}+8 x \,{\mathrm e}^{2 i \sqrt {2}\, x}+2 i \sqrt {2}-8 i x +2 \sqrt {2}-8 x \right ) {\mathrm e}^{2 \sqrt {2}\, x} \end {align*}
\begin {align*} U_2 &= (-1)^{4-2} \int { \frac {F(x) W_2(x) }{a W(x)} \, dx}\\ &= (-1)^{2} \int { \frac { \left (x \,{\mathrm e}^{\sqrt {2}\, x} \sin \left (\sqrt {2}\, x \right )+{\mathrm e}^{-\sqrt {2}\, x} \cos \left (\sqrt {2}\, x \right )\right ) \left (\left (-16+16 i\right ) \sqrt {2}\, {\mathrm e}^{\left (-1+i\right ) \sqrt {2}\, x}\right )}{\left (1\right ) \left (1024\right )} \, dx} \\ &= \int { \frac {\left (-16+16 i\right ) \left (x \,{\mathrm e}^{\sqrt {2}\, x} \sin \left (\sqrt {2}\, x \right )+{\mathrm e}^{-\sqrt {2}\, x} \cos \left (\sqrt {2}\, x \right )\right ) \sqrt {2}\, {\mathrm e}^{\left (-1+i\right ) \sqrt {2}\, x}}{1024} \, dx}\\ &= \int {\left (\left (-\frac {1}{64}+\frac {i}{64}\right ) \sqrt {2}\, {\mathrm e}^{\left (-2+i\right ) \sqrt {2}\, x} \left (x \,{\mathrm e}^{2 \sqrt {2}\, x} \sin \left (\sqrt {2}\, x \right )+\cos \left (\sqrt {2}\, x \right )\right )\right ) \, dx} \\ &= \int \left (-\frac {1}{64}+\frac {i}{64}\right ) \sqrt {2}\, {\mathrm e}^{\left (-2+i\right ) \sqrt {2}\, x} \left (x \,{\mathrm e}^{2 \sqrt {2}\, x} \sin \left (\sqrt {2}\, x \right )+\cos \left (\sqrt {2}\, x \right )\right )d x \end {align*}
\begin {align*} U_3 &= (-1)^{4-3} \int { \frac {F(x) W_3(x) }{a W(x)} \, dx}\\ &= (-1)^{1} \int { \frac { \left (x \,{\mathrm e}^{\sqrt {2}\, x} \sin \left (\sqrt {2}\, x \right )+{\mathrm e}^{-\sqrt {2}\, x} \cos \left (\sqrt {2}\, x \right )\right ) \left (\left (-16+16 i\right ) \sqrt {2}\, {\mathrm e}^{\left (1-i\right ) \sqrt {2}\, x}\right )}{\left (1\right ) \left (1024\right )} \, dx} \\ &= - \int { \frac {\left (-16+16 i\right ) \left (x \,{\mathrm e}^{\sqrt {2}\, x} \sin \left (\sqrt {2}\, x \right )+{\mathrm e}^{-\sqrt {2}\, x} \cos \left (\sqrt {2}\, x \right )\right ) \sqrt {2}\, {\mathrm e}^{\left (1-i\right ) \sqrt {2}\, x}}{1024} \, dx}\\ &= - \int {\left (\left (-\frac {1}{64}+\frac {i}{64}\right ) \sqrt {2}\, {\mathrm e}^{-i \sqrt {2}\, x} \left (x \,{\mathrm e}^{2 \sqrt {2}\, x} \sin \left (\sqrt {2}\, x \right )+\cos \left (\sqrt {2}\, x \right )\right )\right ) \, dx} \\ &= -\left (\int \left (-\frac {1}{64}+\frac {i}{64}\right ) \sqrt {2}\, {\mathrm e}^{-i \sqrt {2}\, x} \left (x \,{\mathrm e}^{2 \sqrt {2}\, x} \sin \left (\sqrt {2}\, x \right )+\cos \left (\sqrt {2}\, x \right )\right )d x \right ) \end {align*}
\begin {align*} U_4 &= (-1)^{4-4} \int { \frac {F(x) W_4(x) }{a W(x)} \, dx}\\ &= (-1)^{0} \int { \frac { \left (x \,{\mathrm e}^{\sqrt {2}\, x} \sin \left (\sqrt {2}\, x \right )+{\mathrm e}^{-\sqrt {2}\, x} \cos \left (\sqrt {2}\, x \right )\right ) \left (\left (-16-16 i\right ) \sqrt {2}\, {\mathrm e}^{\left (-1-i\right ) \sqrt {2}\, x}\right )}{\left (1\right ) \left (1024\right )} \, dx} \\ &= \int { \frac {\left (-16-16 i\right ) \left (x \,{\mathrm e}^{\sqrt {2}\, x} \sin \left (\sqrt {2}\, x \right )+{\mathrm e}^{-\sqrt {2}\, x} \cos \left (\sqrt {2}\, x \right )\right ) \sqrt {2}\, {\mathrm e}^{\left (-1-i\right ) \sqrt {2}\, x}}{1024} \, dx}\\ &= \int {\left (\left (-\frac {1}{64}-\frac {i}{64}\right ) \sqrt {2}\, {\mathrm e}^{\left (-2-i\right ) \sqrt {2}\, x} \left (x \,{\mathrm e}^{2 \sqrt {2}\, x} \sin \left (\sqrt {2}\, x \right )+\cos \left (\sqrt {2}\, x \right )\right )\right ) \, dx} \\ &= \int \left (-\frac {1}{64}-\frac {i}{64}\right ) \sqrt {2}\, {\mathrm e}^{\left (-2-i\right ) \sqrt {2}\, x} \left (x \,{\mathrm e}^{2 \sqrt {2}\, x} \sin \left (\sqrt {2}\, x \right )+\cos \left (\sqrt {2}\, x \right )\right )d x \end {align*}
Now that all the \(U_i\) functions have been determined, the particular solution is found from \[ y_p = U_1 y_1+U_2 y_2+U_3 y_3+U_4 y_4 \] Hence \begin {equation*} \begin {split} y_p &= \left (\frac {\sqrt {2}\, x}{128}+\frac {i \sqrt {2}\, x}{128}+\frac {{\mathrm e}^{2 i \sqrt {2}\, x}}{256}-\frac {i {\mathrm e}^{2 i \sqrt {2}\, x}}{256}+\left (-\frac {1}{8192}-\frac {i}{8192}\right ) \sqrt {2}\, \left (\sqrt {2}+i \sqrt {2}\right ) \left (i \sqrt {2}\, {\mathrm e}^{2 i \sqrt {2}\, x}-{\mathrm e}^{2 i \sqrt {2}\, x} \sqrt {2}+8 x \,{\mathrm e}^{2 i \sqrt {2}\, x}+2 i \sqrt {2}-8 i x +2 \sqrt {2}-8 x \right ) {\mathrm e}^{2 \sqrt {2}\, x}\right ) \left ({\mathrm e}^{\left (-\sqrt {2}-i \sqrt {2}\right ) x}\right ) \\ &+\left (\int \left (-\frac {1}{64}+\frac {i}{64}\right ) \sqrt {2}\, {\mathrm e}^{\left (-2+i\right ) \sqrt {2}\, x} \left (x \,{\mathrm e}^{2 \sqrt {2}\, x} \sin \left (\sqrt {2}\, x \right )+\cos \left (\sqrt {2}\, x \right )\right )d x\right ) \left ({\mathrm e}^{\left (\sqrt {2}-i \sqrt {2}\right ) x}\right ) \\ &+\left (-\left (\int \left (-\frac {1}{64}+\frac {i}{64}\right ) \sqrt {2}\, {\mathrm e}^{-i \sqrt {2}\, x} \left (x \,{\mathrm e}^{2 \sqrt {2}\, x} \sin \left (\sqrt {2}\, x \right )+\cos \left (\sqrt {2}\, x \right )\right )d x \right )\right ) \left ({\mathrm e}^{\left (-\sqrt {2}+i \sqrt {2}\right ) x}\right ) \\ &+\left (\int \left (-\frac {1}{64}-\frac {i}{64}\right ) \sqrt {2}\, {\mathrm e}^{\left (-2-i\right ) \sqrt {2}\, x} \left (x \,{\mathrm e}^{2 \sqrt {2}\, x} \sin \left (\sqrt {2}\, x \right )+\cos \left (\sqrt {2}\, x \right )\right )d x\right ) \left ({\mathrm e}^{\left (\sqrt {2}+i \sqrt {2}\right ) x}\right ) \end {split} \end {equation*} Therefore the particular solution is \[ y_p = \frac {{\mathrm e}^{-\sqrt {2}\, x} \left (\left (4 \sqrt {2}\, x -1+i\right ) \sin \left (\sqrt {2}\, x \right )+\cos \left (\sqrt {2}\, x \right ) \left (4 \sqrt {2}\, x +7+i\right )\right )}{256}-\frac {{\mathrm e}^{\sqrt {2}\, x} \left (\left (\sqrt {2}\, x^{2}-3 x +\frac {7 \sqrt {2}}{8}\right ) \sin \left (\sqrt {2}\, x \right )+\sqrt {2}\, \cos \left (\sqrt {2}\, x \right ) \left (x^{2}-\frac {3}{8}\right )\right )}{128} \] Which simplifies to \[ y_p = \frac {{\mathrm e}^{-\sqrt {2}\, x} \left (\left (4 \sqrt {2}\, x -1+i\right ) \sin \left (\sqrt {2}\, x \right )+\cos \left (\sqrt {2}\, x \right ) \left (4 \sqrt {2}\, x +7+i\right )\right )}{256}-\frac {{\mathrm e}^{\sqrt {2}\, x} \left (\left (\sqrt {2}\, x^{2}-3 x +\frac {7 \sqrt {2}}{8}\right ) \sin \left (\sqrt {2}\, x \right )+\sqrt {2}\, \cos \left (\sqrt {2}\, x \right ) \left (x^{2}-\frac {3}{8}\right )\right )}{128} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left ({\mathrm e}^{\left (-\sqrt {2}-i \sqrt {2}\right ) x} c_{1} +{\mathrm e}^{\left (\sqrt {2}-i \sqrt {2}\right ) x} c_{2} +{\mathrm e}^{\left (-\sqrt {2}+i \sqrt {2}\right ) x} c_{3} +{\mathrm e}^{\left (\sqrt {2}+i \sqrt {2}\right ) x} c_{4}\right ) + \left (\frac {{\mathrm e}^{-\sqrt {2}\, x} \left (\left (4 \sqrt {2}\, x -1+i\right ) \sin \left (\sqrt {2}\, x \right )+\cos \left (\sqrt {2}\, x \right ) \left (4 \sqrt {2}\, x +7+i\right )\right )}{256}-\frac {{\mathrm e}^{\sqrt {2}\, x} \left (\left (\sqrt {2}\, x^{2}-3 x +\frac {7 \sqrt {2}}{8}\right ) \sin \left (\sqrt {2}\, x \right )+\sqrt {2}\, \cos \left (\sqrt {2}\, x \right ) \left (x^{2}-\frac {3}{8}\right )\right )}{128}\right ) \\ \end{align*} Which simplifies to \[ y = {\mathrm e}^{\left (-1-i\right ) \sqrt {2}\, x} c_{1} +{\mathrm e}^{\left (1-i\right ) \sqrt {2}\, x} c_{2} +{\mathrm e}^{\left (-1+i\right ) \sqrt {2}\, x} c_{3} +{\mathrm e}^{\left (1+i\right ) \sqrt {2}\, x} c_{4} +\frac {{\mathrm e}^{-\sqrt {2}\, x} \left (\left (4 \sqrt {2}\, x -1+i\right ) \sin \left (\sqrt {2}\, x \right )+\cos \left (\sqrt {2}\, x \right ) \left (4 \sqrt {2}\, x +7+i\right )\right )}{256}-\frac {{\mathrm e}^{\sqrt {2}\, x} \left (\left (\sqrt {2}\, x^{2}-3 x +\frac {7 \sqrt {2}}{8}\right ) \sin \left (\sqrt {2}\, x \right )+\sqrt {2}\, \cos \left (\sqrt {2}\, x \right ) \left (x^{2}-\frac {3}{8}\right )\right )}{128} \]
The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{\left (-1-i\right ) \sqrt {2}\, x} c_{1} +{\mathrm e}^{\left (1-i\right ) \sqrt {2}\, x} c_{2} +{\mathrm e}^{\left (-1+i\right ) \sqrt {2}\, x} c_{3} +{\mathrm e}^{\left (1+i\right ) \sqrt {2}\, x} c_{4} +\frac {{\mathrm e}^{-\sqrt {2}\, x} \left (\left (4 \sqrt {2}\, x -1+i\right ) \sin \left (\sqrt {2}\, x \right )+\cos \left (\sqrt {2}\, x \right ) \left (4 \sqrt {2}\, x +7+i\right )\right )}{256}-\frac {{\mathrm e}^{\sqrt {2}\, x} \left (\left (\sqrt {2}\, x^{2}-3 x +\frac {7 \sqrt {2}}{8}\right ) \sin \left (\sqrt {2}\, x \right )+\sqrt {2}\, \cos \left (\sqrt {2}\, x \right ) \left (x^{2}-\frac {3}{8}\right )\right )}{128} \\ \end{align*}
Verification of solutions
\[ y = {\mathrm e}^{\left (-1-i\right ) \sqrt {2}\, x} c_{1} +{\mathrm e}^{\left (1-i\right ) \sqrt {2}\, x} c_{2} +{\mathrm e}^{\left (-1+i\right ) \sqrt {2}\, x} c_{3} +{\mathrm e}^{\left (1+i\right ) \sqrt {2}\, x} c_{4} +\frac {{\mathrm e}^{-\sqrt {2}\, x} \left (\left (4 \sqrt {2}\, x -1+i\right ) \sin \left (\sqrt {2}\, x \right )+\cos \left (\sqrt {2}\, x \right ) \left (4 \sqrt {2}\, x +7+i\right )\right )}{256}-\frac {{\mathrm e}^{\sqrt {2}\, x} \left (\left (\sqrt {2}\, x^{2}-3 x +\frac {7 \sqrt {2}}{8}\right ) \sin \left (\sqrt {2}\, x \right )+\sqrt {2}\, \cos \left (\sqrt {2}\, x \right ) \left (x^{2}-\frac {3}{8}\right )\right )}{128} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime \prime \prime \prime }+16 y=x \,{\mathrm e}^{\sqrt {2}\, x} \sin \left (\sqrt {2}\, x \right )+{\mathrm e}^{-\sqrt {2}\, x} \cos \left (\sqrt {2}\, x \right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 4 \\ {} & {} & y^{\prime \prime \prime \prime } \\ \square & {} & \textrm {Convert linear ODE into a system of first order ODEs}\hspace {3pt} \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} y_{1}\left (x \right ) \\ {} & {} & y_{1}\left (x \right )=y \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} y_{2}\left (x \right ) \\ {} & {} & y_{2}\left (x \right )=y^{\prime } \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} y_{3}\left (x \right ) \\ {} & {} & y_{3}\left (x \right )=y^{\prime \prime } \\ {} & \circ & \textrm {Define new variable}\hspace {3pt} y_{4}\left (x \right ) \\ {} & {} & y_{4}\left (x \right )=y^{\prime \prime \prime } \\ {} & \circ & \textrm {Isolate for}\hspace {3pt} y_{4}^{\prime }\left (x \right )\hspace {3pt}\textrm {using original ODE}\hspace {3pt} \\ {} & {} & y_{4}^{\prime }\left (x \right )=x \,{\mathrm e}^{\sqrt {2}\, x} \sin \left (\sqrt {2}\, x \right )+{\mathrm e}^{-\sqrt {2}\, x} \cos \left (\sqrt {2}\, x \right )-16 y_{1}\left (x \right ) \\ & {} & \textrm {Convert linear ODE into a system of first order ODEs}\hspace {3pt} \\ {} & {} & \left [y_{2}\left (x \right )=y_{1}^{\prime }\left (x \right ), y_{3}\left (x \right )=y_{2}^{\prime }\left (x \right ), y_{4}\left (x \right )=y_{3}^{\prime }\left (x \right ), y_{4}^{\prime }\left (x \right )=x \,{\mathrm e}^{\sqrt {2}\, x} \sin \left (\sqrt {2}\, x \right )+{\mathrm e}^{-\sqrt {2}\, x} \cos \left (\sqrt {2}\, x \right )-16 y_{1}\left (x \right )\right ] \\ \bullet & {} & \textrm {Define vector}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}\left (x \right )=\left [\begin {array}{c} y_{1}\left (x \right ) \\ y_{2}\left (x \right ) \\ y_{3}\left (x \right ) \\ y_{4}\left (x \right ) \end {array}\right ] \\ \bullet & {} & \textrm {System to solve}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}^{\prime }\left (x \right )=\left [\begin {array}{cccc} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -16 & 0 & 0 & 0 \end {array}\right ]\cdot {\moverset {\rightarrow }{y}}\left (x \right )+\left [\begin {array}{c} 0 \\ 0 \\ 0 \\ x \,{\mathrm e}^{\sqrt {2}\, x} \sin \left (\sqrt {2}\, x \right )+{\mathrm e}^{-\sqrt {2}\, x} \cos \left (\sqrt {2}\, x \right ) \end {array}\right ] \\ \bullet & {} & \textrm {Define the forcing function}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{f}}\left (x \right )=\left [\begin {array}{c} 0 \\ 0 \\ 0 \\ x \,{\mathrm e}^{\sqrt {2}\, x} \sin \left (\sqrt {2}\, x \right )+{\mathrm e}^{-\sqrt {2}\, x} \cos \left (\sqrt {2}\, x \right ) \end {array}\right ] \\ \bullet & {} & \textrm {Define the coefficient matrix}\hspace {3pt} \\ {} & {} & A =\left [\begin {array}{cccc} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -16 & 0 & 0 & 0 \end {array}\right ] \\ \bullet & {} & \textrm {Rewrite the system as}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}^{\prime }\left (x \right )=A \cdot {\moverset {\rightarrow }{y}}\left (x \right )+{\moverset {\rightarrow }{f}} \\ \bullet & {} & \textrm {To solve the system, find the eigenvalues and eigenvectors of}\hspace {3pt} A \\ \bullet & {} & \textrm {Eigenpairs of}\hspace {3pt} A \\ {} & {} & \left [\left [-\sqrt {2}-\mathrm {I} \sqrt {2}, \left [\begin {array}{c} \frac {1}{\left (-\sqrt {2}-\mathrm {I} \sqrt {2}\right )^{3}} \\ \frac {1}{\left (-\sqrt {2}-\mathrm {I} \sqrt {2}\right )^{2}} \\ \frac {1}{-\sqrt {2}-\mathrm {I} \sqrt {2}} \\ 1 \end {array}\right ]\right ], \left [-\sqrt {2}+\mathrm {I} \sqrt {2}, \left [\begin {array}{c} \frac {1}{\left (-\sqrt {2}+\mathrm {I} \sqrt {2}\right )^{3}} \\ \frac {1}{\left (-\sqrt {2}+\mathrm {I} \sqrt {2}\right )^{2}} \\ \frac {1}{-\sqrt {2}+\mathrm {I} \sqrt {2}} \\ 1 \end {array}\right ]\right ], \left [\sqrt {2}-\mathrm {I} \sqrt {2}, \left [\begin {array}{c} \frac {1}{\left (\sqrt {2}-\mathrm {I} \sqrt {2}\right )^{3}} \\ \frac {1}{\left (\sqrt {2}-\mathrm {I} \sqrt {2}\right )^{2}} \\ \frac {1}{\sqrt {2}-\mathrm {I} \sqrt {2}} \\ 1 \end {array}\right ]\right ], \left [\sqrt {2}+\mathrm {I} \sqrt {2}, \left [\begin {array}{c} \frac {1}{\left (\sqrt {2}+\mathrm {I} \sqrt {2}\right )^{3}} \\ \frac {1}{\left (\sqrt {2}+\mathrm {I} \sqrt {2}\right )^{2}} \\ \frac {1}{\sqrt {2}+\mathrm {I} \sqrt {2}} \\ 1 \end {array}\right ]\right ]\right ] \\ \bullet & {} & \textrm {Consider complex eigenpair, complex conjugate eigenvalue can be ignored}\hspace {3pt} \\ {} & {} & \left [-\sqrt {2}-\mathrm {I} \sqrt {2}, \left [\begin {array}{c} \frac {1}{\left (-\sqrt {2}-\mathrm {I} \sqrt {2}\right )^{3}} \\ \frac {1}{\left (-\sqrt {2}-\mathrm {I} \sqrt {2}\right )^{2}} \\ \frac {1}{-\sqrt {2}-\mathrm {I} \sqrt {2}} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution from eigenpair}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (-\sqrt {2}-\mathrm {I} \sqrt {2}\right ) x}\cdot \left [\begin {array}{c} \frac {1}{\left (-\sqrt {2}-\mathrm {I} \sqrt {2}\right )^{3}} \\ \frac {1}{\left (-\sqrt {2}-\mathrm {I} \sqrt {2}\right )^{2}} \\ \frac {1}{-\sqrt {2}-\mathrm {I} \sqrt {2}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Use Euler identity to write solution in terms of}\hspace {3pt} \sin \hspace {3pt}\textrm {and}\hspace {3pt} \cos \\ {} & {} & {\mathrm e}^{-\sqrt {2}\, x}\cdot \left (\cos \left (\sqrt {2}\, x \right )-\mathrm {I} \sin \left (\sqrt {2}\, x \right )\right )\cdot \left [\begin {array}{c} \frac {1}{\left (-\sqrt {2}-\mathrm {I} \sqrt {2}\right )^{3}} \\ \frac {1}{\left (-\sqrt {2}-\mathrm {I} \sqrt {2}\right )^{2}} \\ \frac {1}{-\sqrt {2}-\mathrm {I} \sqrt {2}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Simplify expression}\hspace {3pt} \\ {} & {} & {\mathrm e}^{-\sqrt {2}\, x}\cdot \left [\begin {array}{c} \frac {\cos \left (\sqrt {2}\, x \right )-\mathrm {I} \sin \left (\sqrt {2}\, x \right )}{\left (-\sqrt {2}-\mathrm {I} \sqrt {2}\right )^{3}} \\ \frac {\cos \left (\sqrt {2}\, x \right )-\mathrm {I} \sin \left (\sqrt {2}\, x \right )}{\left (-\sqrt {2}-\mathrm {I} \sqrt {2}\right )^{2}} \\ \frac {\cos \left (\sqrt {2}\, x \right )-\mathrm {I} \sin \left (\sqrt {2}\, x \right )}{-\sqrt {2}-\mathrm {I} \sqrt {2}} \\ \cos \left (\sqrt {2}\, x \right )-\mathrm {I} \sin \left (\sqrt {2}\, x \right ) \end {array}\right ] \\ \bullet & {} & \textrm {Both real and imaginary parts are solutions to the homogeneous system}\hspace {3pt} \\ {} & {} & \left [{\moverset {\rightarrow }{y}}_{1}\left (x \right )={\mathrm e}^{-\sqrt {2}\, x}\cdot \left [\begin {array}{c} \frac {\cos \left (\sqrt {2}\, x \right ) \sqrt {2}}{16}+\frac {\sin \left (\sqrt {2}\, x \right ) \sqrt {2}}{16} \\ -\frac {\sin \left (\sqrt {2}\, x \right )}{4} \\ -\frac {\cos \left (\sqrt {2}\, x \right ) \sqrt {2}}{4}+\frac {\sin \left (\sqrt {2}\, x \right ) \sqrt {2}}{4} \\ \cos \left (\sqrt {2}\, x \right ) \end {array}\right ], {\moverset {\rightarrow }{y}}_{2}\left (x \right )={\mathrm e}^{-\sqrt {2}\, x}\cdot \left [\begin {array}{c} \frac {\cos \left (\sqrt {2}\, x \right ) \sqrt {2}}{16}-\frac {\sin \left (\sqrt {2}\, x \right ) \sqrt {2}}{16} \\ -\frac {\cos \left (\sqrt {2}\, x \right )}{4} \\ \frac {\cos \left (\sqrt {2}\, x \right ) \sqrt {2}}{4}+\frac {\sin \left (\sqrt {2}\, x \right ) \sqrt {2}}{4} \\ -\sin \left (\sqrt {2}\, x \right ) \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Consider complex eigenpair, complex conjugate eigenvalue can be ignored}\hspace {3pt} \\ {} & {} & \left [\sqrt {2}-\mathrm {I} \sqrt {2}, \left [\begin {array}{c} \frac {1}{\left (\sqrt {2}-\mathrm {I} \sqrt {2}\right )^{3}} \\ \frac {1}{\left (\sqrt {2}-\mathrm {I} \sqrt {2}\right )^{2}} \\ \frac {1}{\sqrt {2}-\mathrm {I} \sqrt {2}} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution from eigenpair}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (\sqrt {2}-\mathrm {I} \sqrt {2}\right ) x}\cdot \left [\begin {array}{c} \frac {1}{\left (\sqrt {2}-\mathrm {I} \sqrt {2}\right )^{3}} \\ \frac {1}{\left (\sqrt {2}-\mathrm {I} \sqrt {2}\right )^{2}} \\ \frac {1}{\sqrt {2}-\mathrm {I} \sqrt {2}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Use Euler identity to write solution in terms of}\hspace {3pt} \sin \hspace {3pt}\textrm {and}\hspace {3pt} \cos \\ {} & {} & {\mathrm e}^{\sqrt {2}\, x}\cdot \left (\cos \left (\sqrt {2}\, x \right )-\mathrm {I} \sin \left (\sqrt {2}\, x \right )\right )\cdot \left [\begin {array}{c} \frac {1}{\left (\sqrt {2}-\mathrm {I} \sqrt {2}\right )^{3}} \\ \frac {1}{\left (\sqrt {2}-\mathrm {I} \sqrt {2}\right )^{2}} \\ \frac {1}{\sqrt {2}-\mathrm {I} \sqrt {2}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Simplify expression}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\sqrt {2}\, x}\cdot \left [\begin {array}{c} \frac {\cos \left (\sqrt {2}\, x \right )-\mathrm {I} \sin \left (\sqrt {2}\, x \right )}{\left (\sqrt {2}-\mathrm {I} \sqrt {2}\right )^{3}} \\ \frac {\cos \left (\sqrt {2}\, x \right )-\mathrm {I} \sin \left (\sqrt {2}\, x \right )}{\left (\sqrt {2}-\mathrm {I} \sqrt {2}\right )^{2}} \\ \frac {\cos \left (\sqrt {2}\, x \right )-\mathrm {I} \sin \left (\sqrt {2}\, x \right )}{\sqrt {2}-\mathrm {I} \sqrt {2}} \\ \cos \left (\sqrt {2}\, x \right )-\mathrm {I} \sin \left (\sqrt {2}\, x \right ) \end {array}\right ] \\ \bullet & {} & \textrm {Both real and imaginary parts are solutions to the homogeneous system}\hspace {3pt} \\ {} & {} & \left [{\moverset {\rightarrow }{y}}_{3}\left (x \right )={\mathrm e}^{\sqrt {2}\, x}\cdot \left [\begin {array}{c} -\frac {\cos \left (\sqrt {2}\, x \right ) \sqrt {2}}{16}+\frac {\sin \left (\sqrt {2}\, x \right ) \sqrt {2}}{16} \\ \frac {\sin \left (\sqrt {2}\, x \right )}{4} \\ \frac {\cos \left (\sqrt {2}\, x \right ) \sqrt {2}}{4}+\frac {\sin \left (\sqrt {2}\, x \right ) \sqrt {2}}{4} \\ \cos \left (\sqrt {2}\, x \right ) \end {array}\right ], {\moverset {\rightarrow }{y}}_{4}\left (x \right )={\mathrm e}^{\sqrt {2}\, x}\cdot \left [\begin {array}{c} \frac {\cos \left (\sqrt {2}\, x \right ) \sqrt {2}}{16}+\frac {\sin \left (\sqrt {2}\, x \right ) \sqrt {2}}{16} \\ \frac {\cos \left (\sqrt {2}\, x \right )}{4} \\ \frac {\cos \left (\sqrt {2}\, x \right ) \sqrt {2}}{4}-\frac {\sin \left (\sqrt {2}\, x \right ) \sqrt {2}}{4} \\ -\sin \left (\sqrt {2}\, x \right ) \end {array}\right ]\right ] \\ \bullet & {} & \textrm {General solution of the system of ODEs can be written in terms of the particular solution}\hspace {3pt} {\moverset {\rightarrow }{y}}_{p}\left (x \right ) \\ {} & {} & {\moverset {\rightarrow }{y}}\left (x \right )=c_{1} {\moverset {\rightarrow }{y}}_{1}\left (x \right )+c_{2} {\moverset {\rightarrow }{y}}_{2}\left (x \right )+c_{3} {\moverset {\rightarrow }{y}}_{3}\left (x \right )+c_{4} {\moverset {\rightarrow }{y}}_{4}\left (x \right )+{\moverset {\rightarrow }{y}}_{p}\left (x \right ) \\ \square & {} & \textrm {Fundamental matrix}\hspace {3pt} \\ {} & \circ & \textrm {Let}\hspace {3pt} \phi \left (x \right )\hspace {3pt}\textrm {be the matrix whose columns are the independent solutions of the homogeneous system.}\hspace {3pt} \\ {} & {} & \phi \left (x \right )=\left [\begin {array}{cccc} {\mathrm e}^{-\sqrt {2}\, x} \left (\frac {\cos \left (\sqrt {2}\, x \right ) \sqrt {2}}{16}+\frac {\sin \left (\sqrt {2}\, x \right ) \sqrt {2}}{16}\right ) & {\mathrm e}^{-\sqrt {2}\, x} \left (\frac {\cos \left (\sqrt {2}\, x \right ) \sqrt {2}}{16}-\frac {\sin \left (\sqrt {2}\, x \right ) \sqrt {2}}{16}\right ) & {\mathrm e}^{\sqrt {2}\, x} \left (-\frac {\cos \left (\sqrt {2}\, x \right ) \sqrt {2}}{16}+\frac {\sin \left (\sqrt {2}\, x \right ) \sqrt {2}}{16}\right ) & {\mathrm e}^{\sqrt {2}\, x} \left (\frac {\cos \left (\sqrt {2}\, x \right ) \sqrt {2}}{16}+\frac {\sin \left (\sqrt {2}\, x \right ) \sqrt {2}}{16}\right ) \\ -\frac {{\mathrm e}^{-\sqrt {2}\, x} \sin \left (\sqrt {2}\, x \right )}{4} & -\frac {{\mathrm e}^{-\sqrt {2}\, x} \cos \left (\sqrt {2}\, x \right )}{4} & \frac {{\mathrm e}^{\sqrt {2}\, x} \sin \left (\sqrt {2}\, x \right )}{4} & \frac {{\mathrm e}^{\sqrt {2}\, x} \cos \left (\sqrt {2}\, x \right )}{4} \\ {\mathrm e}^{-\sqrt {2}\, x} \left (-\frac {\cos \left (\sqrt {2}\, x \right ) \sqrt {2}}{4}+\frac {\sin \left (\sqrt {2}\, x \right ) \sqrt {2}}{4}\right ) & {\mathrm e}^{-\sqrt {2}\, x} \left (\frac {\cos \left (\sqrt {2}\, x \right ) \sqrt {2}}{4}+\frac {\sin \left (\sqrt {2}\, x \right ) \sqrt {2}}{4}\right ) & {\mathrm e}^{\sqrt {2}\, x} \left (\frac {\cos \left (\sqrt {2}\, x \right ) \sqrt {2}}{4}+\frac {\sin \left (\sqrt {2}\, x \right ) \sqrt {2}}{4}\right ) & {\mathrm e}^{\sqrt {2}\, x} \left (\frac {\cos \left (\sqrt {2}\, x \right ) \sqrt {2}}{4}-\frac {\sin \left (\sqrt {2}\, x \right ) \sqrt {2}}{4}\right ) \\ {\mathrm e}^{-\sqrt {2}\, x} \cos \left (\sqrt {2}\, x \right ) & -{\mathrm e}^{-\sqrt {2}\, x} \sin \left (\sqrt {2}\, x \right ) & {\mathrm e}^{\sqrt {2}\, x} \cos \left (\sqrt {2}\, x \right ) & -{\mathrm e}^{\sqrt {2}\, x} \sin \left (\sqrt {2}\, x \right ) \end {array}\right ] \\ {} & \circ & \textrm {The fundamental matrix,}\hspace {3pt} \Phi \left (x \right )\hspace {3pt}\textrm {is a normalized version of}\hspace {3pt} \phi \left (x \right )\hspace {3pt}\textrm {satisfying}\hspace {3pt} \Phi \left (0\right )=I \hspace {3pt}\textrm {where}\hspace {3pt} I \hspace {3pt}\textrm {is the identity matrix}\hspace {3pt} \\ {} & {} & \Phi \left (x \right )=\phi \left (x \right )\cdot \frac {1}{\phi \left (0\right )} \\ {} & \circ & \textrm {Substitute the value of}\hspace {3pt} \phi \left (x \right )\hspace {3pt}\textrm {and}\hspace {3pt} \phi \left (0\right ) \\ {} & {} & \Phi \left (x \right )=\left [\begin {array}{cccc} {\mathrm e}^{-\sqrt {2}\, x} \left (\frac {\cos \left (\sqrt {2}\, x \right ) \sqrt {2}}{16}+\frac {\sin \left (\sqrt {2}\, x \right ) \sqrt {2}}{16}\right ) & {\mathrm e}^{-\sqrt {2}\, x} \left (\frac {\cos \left (\sqrt {2}\, x \right ) \sqrt {2}}{16}-\frac {\sin \left (\sqrt {2}\, x \right ) \sqrt {2}}{16}\right ) & {\mathrm e}^{\sqrt {2}\, x} \left (-\frac {\cos \left (\sqrt {2}\, x \right ) \sqrt {2}}{16}+\frac {\sin \left (\sqrt {2}\, x \right ) \sqrt {2}}{16}\right ) & {\mathrm e}^{\sqrt {2}\, x} \left (\frac {\cos \left (\sqrt {2}\, x \right ) \sqrt {2}}{16}+\frac {\sin \left (\sqrt {2}\, x \right ) \sqrt {2}}{16}\right ) \\ -\frac {{\mathrm e}^{-\sqrt {2}\, x} \sin \left (\sqrt {2}\, x \right )}{4} & -\frac {{\mathrm e}^{-\sqrt {2}\, x} \cos \left (\sqrt {2}\, x \right )}{4} & \frac {{\mathrm e}^{\sqrt {2}\, x} \sin \left (\sqrt {2}\, x \right )}{4} & \frac {{\mathrm e}^{\sqrt {2}\, x} \cos \left (\sqrt {2}\, x \right )}{4} \\ {\mathrm e}^{-\sqrt {2}\, x} \left (-\frac {\cos \left (\sqrt {2}\, x \right ) \sqrt {2}}{4}+\frac {\sin \left (\sqrt {2}\, x \right ) \sqrt {2}}{4}\right ) & {\mathrm e}^{-\sqrt {2}\, x} \left (\frac {\cos \left (\sqrt {2}\, x \right ) \sqrt {2}}{4}+\frac {\sin \left (\sqrt {2}\, x \right ) \sqrt {2}}{4}\right ) & {\mathrm e}^{\sqrt {2}\, x} \left (\frac {\cos \left (\sqrt {2}\, x \right ) \sqrt {2}}{4}+\frac {\sin \left (\sqrt {2}\, x \right ) \sqrt {2}}{4}\right ) & {\mathrm e}^{\sqrt {2}\, x} \left (\frac {\cos \left (\sqrt {2}\, x \right ) \sqrt {2}}{4}-\frac {\sin \left (\sqrt {2}\, x \right ) \sqrt {2}}{4}\right ) \\ {\mathrm e}^{-\sqrt {2}\, x} \cos \left (\sqrt {2}\, x \right ) & -{\mathrm e}^{-\sqrt {2}\, x} \sin \left (\sqrt {2}\, x \right ) & {\mathrm e}^{\sqrt {2}\, x} \cos \left (\sqrt {2}\, x \right ) & -{\mathrm e}^{\sqrt {2}\, x} \sin \left (\sqrt {2}\, x \right ) \end {array}\right ]\cdot \frac {1}{\left [\begin {array}{cccc} \frac {\sqrt {2}}{16} & \frac {\sqrt {2}}{16} & -\frac {\sqrt {2}}{16} & \frac {\sqrt {2}}{16} \\ 0 & -\frac {1}{4} & 0 & \frac {1}{4} \\ -\frac {\sqrt {2}}{4} & \frac {\sqrt {2}}{4} & \frac {\sqrt {2}}{4} & \frac {\sqrt {2}}{4} \\ 1 & 0 & 1 & 0 \end {array}\right ]} \\ {} & \circ & \textrm {Evaluate and simplify to get the fundamental matrix}\hspace {3pt} \\ {} & {} & \Phi \left (x \right )=\left [\begin {array}{cccc} \frac {\cos \left (\sqrt {2}\, x \right ) \left ({\mathrm e}^{-\sqrt {2}\, x}+{\mathrm e}^{\sqrt {2}\, x}\right )}{2} & \frac {\sqrt {2}\, \left ({\mathrm e}^{-\sqrt {2}\, x} \left (-\cos \left (\sqrt {2}\, x \right )+\sin \left (\sqrt {2}\, x \right )\right )+{\mathrm e}^{\sqrt {2}\, x} \left (\cos \left (\sqrt {2}\, x \right )+\sin \left (\sqrt {2}\, x \right )\right )\right )}{8} & \frac {\sin \left (\sqrt {2}\, x \right ) \left (-{\mathrm e}^{-\sqrt {2}\, x}+{\mathrm e}^{\sqrt {2}\, x}\right )}{8} & -\frac {\left (\left (-\cos \left (\sqrt {2}\, x \right )-\sin \left (\sqrt {2}\, x \right )\right ) {\mathrm e}^{-\sqrt {2}\, x}+{\mathrm e}^{\sqrt {2}\, x} \left (\cos \left (\sqrt {2}\, x \right )-\sin \left (\sqrt {2}\, x \right )\right )\right ) \sqrt {2}}{32} \\ \frac {\left (\left (-\cos \left (\sqrt {2}\, x \right )-\sin \left (\sqrt {2}\, x \right )\right ) {\mathrm e}^{-\sqrt {2}\, x}+{\mathrm e}^{\sqrt {2}\, x} \left (\cos \left (\sqrt {2}\, x \right )-\sin \left (\sqrt {2}\, x \right )\right )\right ) \sqrt {2}}{2} & \frac {\cos \left (\sqrt {2}\, x \right ) \left ({\mathrm e}^{-\sqrt {2}\, x}+{\mathrm e}^{\sqrt {2}\, x}\right )}{2} & \frac {\sqrt {2}\, \left ({\mathrm e}^{-\sqrt {2}\, x} \left (-\cos \left (\sqrt {2}\, x \right )+\sin \left (\sqrt {2}\, x \right )\right )+{\mathrm e}^{\sqrt {2}\, x} \left (\cos \left (\sqrt {2}\, x \right )+\sin \left (\sqrt {2}\, x \right )\right )\right )}{8} & \frac {\sin \left (\sqrt {2}\, x \right ) \left (-{\mathrm e}^{-\sqrt {2}\, x}+{\mathrm e}^{\sqrt {2}\, x}\right )}{8} \\ -2 \sin \left (\sqrt {2}\, x \right ) \left (-{\mathrm e}^{-\sqrt {2}\, x}+{\mathrm e}^{\sqrt {2}\, x}\right ) & \frac {\left (\left (-\cos \left (\sqrt {2}\, x \right )-\sin \left (\sqrt {2}\, x \right )\right ) {\mathrm e}^{-\sqrt {2}\, x}+{\mathrm e}^{\sqrt {2}\, x} \left (\cos \left (\sqrt {2}\, x \right )-\sin \left (\sqrt {2}\, x \right )\right )\right ) \sqrt {2}}{2} & \frac {\cos \left (\sqrt {2}\, x \right ) \left ({\mathrm e}^{-\sqrt {2}\, x}+{\mathrm e}^{\sqrt {2}\, x}\right )}{2} & \frac {\sqrt {2}\, \left ({\mathrm e}^{-\sqrt {2}\, x} \left (-\cos \left (\sqrt {2}\, x \right )+\sin \left (\sqrt {2}\, x \right )\right )+{\mathrm e}^{\sqrt {2}\, x} \left (\cos \left (\sqrt {2}\, x \right )+\sin \left (\sqrt {2}\, x \right )\right )\right )}{8} \\ -2 \sqrt {2}\, \left ({\mathrm e}^{-\sqrt {2}\, x} \left (-\cos \left (\sqrt {2}\, x \right )+\sin \left (\sqrt {2}\, x \right )\right )+{\mathrm e}^{\sqrt {2}\, x} \left (\cos \left (\sqrt {2}\, x \right )+\sin \left (\sqrt {2}\, x \right )\right )\right ) & -2 \sin \left (\sqrt {2}\, x \right ) \left (-{\mathrm e}^{-\sqrt {2}\, x}+{\mathrm e}^{\sqrt {2}\, x}\right ) & \frac {\left (\left (-\cos \left (\sqrt {2}\, x \right )-\sin \left (\sqrt {2}\, x \right )\right ) {\mathrm e}^{-\sqrt {2}\, x}+{\mathrm e}^{\sqrt {2}\, x} \left (\cos \left (\sqrt {2}\, x \right )-\sin \left (\sqrt {2}\, x \right )\right )\right ) \sqrt {2}}{2} & \frac {\cos \left (\sqrt {2}\, x \right ) \left ({\mathrm e}^{-\sqrt {2}\, x}+{\mathrm e}^{\sqrt {2}\, x}\right )}{2} \end {array}\right ] \\ \square & {} & \textrm {Find a particular solution of the system of ODEs using variation of parameters}\hspace {3pt} \\ {} & \circ & \textrm {Let the particular solution be the fundamental matrix multiplied by}\hspace {3pt} {\moverset {\rightarrow }{v}}\left (x \right )\hspace {3pt}\textrm {and solve for}\hspace {3pt} {\moverset {\rightarrow }{v}}\left (x \right ) \\ {} & {} & {\moverset {\rightarrow }{y}}_{p}\left (x \right )=\Phi \left (x \right )\cdot {\moverset {\rightarrow }{v}}\left (x \right ) \\ {} & \circ & \textrm {Take the derivative of the particular solution}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{p}^{\prime }\left (x \right )=\Phi ^{\prime }\left (x \right )\cdot {\moverset {\rightarrow }{v}}\left (x \right )+\Phi \left (x \right )\cdot {\moverset {\rightarrow }{v}}^{\prime }\left (x \right ) \\ {} & \circ & \textrm {Substitute particular solution and its derivative into the system of ODEs}\hspace {3pt} \\ {} & {} & \Phi ^{\prime }\left (x \right )\cdot {\moverset {\rightarrow }{v}}\left (x \right )+\Phi \left (x \right )\cdot {\moverset {\rightarrow }{v}}^{\prime }\left (x \right )=A \cdot \Phi \left (x \right )\cdot {\moverset {\rightarrow }{v}}\left (x \right )+{\moverset {\rightarrow }{f}}\left (x \right ) \\ {} & \circ & \textrm {The fundamental matrix has columns that are solutions to the homogeneous system so its derivative follows that of the homogeneous system}\hspace {3pt} \\ {} & {} & A \cdot \Phi \left (x \right )\cdot {\moverset {\rightarrow }{v}}\left (x \right )+\Phi \left (x \right )\cdot {\moverset {\rightarrow }{v}}^{\prime }\left (x \right )=A \cdot \Phi \left (x \right )\cdot {\moverset {\rightarrow }{v}}\left (x \right )+{\moverset {\rightarrow }{f}}\left (x \right ) \\ {} & \circ & \textrm {Cancel like terms}\hspace {3pt} \\ {} & {} & \Phi \left (x \right )\cdot {\moverset {\rightarrow }{v}}^{\prime }\left (x \right )={\moverset {\rightarrow }{f}}\left (x \right ) \\ {} & \circ & \textrm {Multiply by the inverse of the fundamental matrix}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{v}}^{\prime }\left (x \right )=\frac {1}{\Phi \left (x \right )}\cdot {\moverset {\rightarrow }{f}}\left (x \right ) \\ {} & \circ & \textrm {Integrate to solve for}\hspace {3pt} {\moverset {\rightarrow }{v}}\left (x \right ) \\ {} & {} & {\moverset {\rightarrow }{v}}\left (x \right )=\int _{0}^{x}\frac {1}{\Phi \left (s \right )}\cdot {\moverset {\rightarrow }{f}}\left (s \right )d s \\ {} & \circ & \textrm {Plug}\hspace {3pt} {\moverset {\rightarrow }{v}}\left (x \right )\hspace {3pt}\textrm {into the equation for the particular solution}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{p}\left (x \right )=\Phi \left (x \right )\cdot \left (\int _{0}^{x}\frac {1}{\Phi \left (s \right )}\cdot {\moverset {\rightarrow }{f}}\left (s \right )d s \right ) \\ {} & \circ & \textrm {Plug in the fundamental matrix and the forcing function and compute}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{p}\left (x \right )=\left [\begin {array}{c} \frac {\left (\left (8+\left (16 x +1\right ) \sqrt {2}\right ) \sin \left (\sqrt {2}\, x \right )+16 \cos \left (\sqrt {2}\, x \right ) \left (1+\left (x -\frac {3}{16}\right ) \sqrt {2}\right )\right ) {\mathrm e}^{-\sqrt {2}\, x}}{1024}-\frac {{\mathrm e}^{\sqrt {2}\, x} \left (\left (\sqrt {2}\, x^{2}-3 x +\frac {7 \sqrt {2}}{8}-1\right ) \sin \left (\sqrt {2}\, x \right )+\cos \left (\sqrt {2}\, x \right ) \left (\sqrt {2}\, x^{2}-\frac {3 \sqrt {2}}{8}+2\right )\right )}{128} \\ -\frac {\left (\left (\left (\frac {1}{4}+\frac {\left (1-x \right ) \sqrt {2}}{4}+x^{2}\right ) \cos \left (\sqrt {2}\, x \right )-\frac {\sin \left (\sqrt {2}\, x \right ) \left (\frac {1}{2}+\left (x +3\right ) \sqrt {2}\right )}{4}\right ) {\mathrm e}^{2 \sqrt {2}\, x}+\frac {\left (-1-\sqrt {2}\right ) \cos \left (\sqrt {2}\, x \right )}{4}+2 \sin \left (\sqrt {2}\, x \right ) \left (x +\frac {\sqrt {2}}{8}-\frac {1}{16}\right )\right ) {\mathrm e}^{-\sqrt {2}\, x}}{32} \\ \frac {\left (\left (-8+\left (-16 x -1\right ) \sqrt {2}\right ) \cos \left (\sqrt {2}\, x \right )+16 \left (-1+\left (x -\frac {3}{16}\right ) \sqrt {2}\right ) \sin \left (\sqrt {2}\, x \right )\right ) {\mathrm e}^{-\sqrt {2}\, x}}{256}-\frac {{\mathrm e}^{\sqrt {2}\, x} \left (\left (\sqrt {2}\, x^{2}+x -\frac {\sqrt {2}}{8}-1\right ) \cos \left (\sqrt {2}\, x \right )-\left (\sqrt {2}\, x^{2}+\frac {5 \sqrt {2}}{8}+2\right ) \sin \left (\sqrt {2}\, x \right )\right )}{32} \\ \frac {\left (\left (16 x -6 \sqrt {2}-1\right ) \cos \left (\sqrt {2}\, x \right )+\left (10 \sqrt {2}+2\right ) \sin \left (\sqrt {2}\, x \right )\right ) {\mathrm e}^{-\sqrt {2}\, x}}{64}+\frac {{\mathrm e}^{\sqrt {2}\, x} \left (\left (\frac {1}{8}+\left (-\frac {3 x}{4}+\frac {3}{4}\right ) \sqrt {2}\right ) \cos \left (\sqrt {2}\, x \right )+\left (\left (\frac {3 x}{4}+\frac {1}{4}\right ) \sqrt {2}+x^{2}+\frac {1}{4}\right ) \sin \left (\sqrt {2}\, x \right )\right )}{8} \end {array}\right ] \\ \bullet & {} & \textrm {Plug particular solution back into general solution}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}\left (x \right )=c_{1} {\moverset {\rightarrow }{y}}_{1}\left (x \right )+c_{2} {\moverset {\rightarrow }{y}}_{2}\left (x \right )+c_{3} {\moverset {\rightarrow }{y}}_{3}\left (x \right )+c_{4} {\moverset {\rightarrow }{y}}_{4}\left (x \right )+\left [\begin {array}{c} \frac {\left (\left (8+\left (16 x +1\right ) \sqrt {2}\right ) \sin \left (\sqrt {2}\, x \right )+16 \cos \left (\sqrt {2}\, x \right ) \left (1+\left (x -\frac {3}{16}\right ) \sqrt {2}\right )\right ) {\mathrm e}^{-\sqrt {2}\, x}}{1024}-\frac {{\mathrm e}^{\sqrt {2}\, x} \left (\left (\sqrt {2}\, x^{2}-3 x +\frac {7 \sqrt {2}}{8}-1\right ) \sin \left (\sqrt {2}\, x \right )+\cos \left (\sqrt {2}\, x \right ) \left (\sqrt {2}\, x^{2}-\frac {3 \sqrt {2}}{8}+2\right )\right )}{128} \\ -\frac {\left (\left (\left (\frac {1}{4}+\frac {\left (1-x \right ) \sqrt {2}}{4}+x^{2}\right ) \cos \left (\sqrt {2}\, x \right )-\frac {\sin \left (\sqrt {2}\, x \right ) \left (\frac {1}{2}+\left (x +3\right ) \sqrt {2}\right )}{4}\right ) {\mathrm e}^{2 \sqrt {2}\, x}+\frac {\left (-1-\sqrt {2}\right ) \cos \left (\sqrt {2}\, x \right )}{4}+2 \sin \left (\sqrt {2}\, x \right ) \left (x +\frac {\sqrt {2}}{8}-\frac {1}{16}\right )\right ) {\mathrm e}^{-\sqrt {2}\, x}}{32} \\ \frac {\left (\left (-8+\left (-16 x -1\right ) \sqrt {2}\right ) \cos \left (\sqrt {2}\, x \right )+16 \left (-1+\left (x -\frac {3}{16}\right ) \sqrt {2}\right ) \sin \left (\sqrt {2}\, x \right )\right ) {\mathrm e}^{-\sqrt {2}\, x}}{256}-\frac {{\mathrm e}^{\sqrt {2}\, x} \left (\left (\sqrt {2}\, x^{2}+x -\frac {\sqrt {2}}{8}-1\right ) \cos \left (\sqrt {2}\, x \right )-\left (\sqrt {2}\, x^{2}+\frac {5 \sqrt {2}}{8}+2\right ) \sin \left (\sqrt {2}\, x \right )\right )}{32} \\ \frac {\left (\left (16 x -6 \sqrt {2}-1\right ) \cos \left (\sqrt {2}\, x \right )+\left (10 \sqrt {2}+2\right ) \sin \left (\sqrt {2}\, x \right )\right ) {\mathrm e}^{-\sqrt {2}\, x}}{64}+\frac {{\mathrm e}^{\sqrt {2}\, x} \left (\left (\frac {1}{8}+\left (-\frac {3 x}{4}+\frac {3}{4}\right ) \sqrt {2}\right ) \cos \left (\sqrt {2}\, x \right )+\left (\left (\frac {3 x}{4}+\frac {1}{4}\right ) \sqrt {2}+x^{2}+\frac {1}{4}\right ) \sin \left (\sqrt {2}\, x \right )\right )}{8} \end {array}\right ] \\ \bullet & {} & \textrm {First component of the vector is the solution to the ODE}\hspace {3pt} \\ {} & {} & y=\frac {\left (\left (8+\left (16 x +64 c_{1} -64 c_{2} +1\right ) \sqrt {2}\right ) \sin \left (\sqrt {2}\, x \right )+16 \left (1+\left (x +4 c_{1} +4 c_{2} -\frac {3}{16}\right ) \sqrt {2}\right ) \cos \left (\sqrt {2}\, x \right )\right ) {\mathrm e}^{-\sqrt {2}\, x}}{1024}-\frac {\left (\left (\left (x^{2}-8 c_{3} -8 c_{4} +\frac {7}{8}\right ) \sqrt {2}-3 x -1\right ) \sin \left (\sqrt {2}\, x \right )+\left (2+\left (x^{2}+8 c_{3} -8 c_{4} -\frac {3}{8}\right ) \sqrt {2}\right ) \cos \left (\sqrt {2}\, x \right )\right ) {\mathrm e}^{\sqrt {2}\, x}}{128} \end {array} \]
Maple trace
`Methods for high order ODEs: --- Trying classification methods --- trying a quadrature trying high order exact linear fully integrable trying differential order: 4; linear nonhomogeneous with symmetry [0,1] trying high order linear exact nonhomogeneous trying differential order: 4; missing the dependent variable checking if the LODE has constant coefficients <- constant coefficients successful`
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 108
dsolve(diff(y(x),x$4)+16*y(x)=x*exp(sqrt(2)*x)*sin(sqrt(2)*x)+exp(-sqrt(2)*x)*cos(sqrt(2)*x),y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (\left (2 x \sqrt {2}+128 c_{3} +3\right ) \cos \left (x \sqrt {2}\right )+2 \sin \left (x \sqrt {2}\right ) \left (x \sqrt {2}+64 c_{4} \right )\right ) {\mathrm e}^{-x \sqrt {2}}}{128}-\frac {\left (\left (x^{2} \sqrt {2}-128 c_{1} -\frac {5 \sqrt {2}}{8}\right ) \cos \left (x \sqrt {2}\right )+\sin \left (x \sqrt {2}\right ) \left (x^{2} \sqrt {2}-3 x -128 c_{2} +\frac {5 \sqrt {2}}{8}\right )\right ) {\mathrm e}^{x \sqrt {2}}}{128} \]
✓ Solution by Mathematica
Time used: 2.857 (sec). Leaf size: 140
DSolve[y''''[x]+16*y[x]==x*Exp[Sqrt[2]*x]*Sin[Sqrt[2]*x]+Exp[-Sqrt[2]*x]*Cos[Sqrt[2]*x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {e^{-\sqrt {2} x} \left (\left (e^{2 \sqrt {2} x} \left (-8 \sqrt {2} x^2+5 \sqrt {2}+1024 c_1\right )+8 \left (2 \sqrt {2} x+3+128 c_2\right )\right ) \cos \left (\sqrt {2} x\right )-\left (e^{2 \sqrt {2} x} \left (8 \sqrt {2} x^2-24 x+5 \sqrt {2}-1024 c_4\right )-16 \left (\sqrt {2} x+64 c_3\right )\right ) \sin \left (\sqrt {2} x\right )\right )}{1024} \]