19.18 problem section 9.3, problem 18

Internal problem ID [1515]
Internal file name [OUTPUT/1516_Sunday_June_05_2022_02_20_19_AM_9905438/index.tex]

Book: Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section: Chapter 9 Introduction to Linear Higher Order Equations. Section 9.3. Undetermined Coefficients for Higher Order Equations. Page 495
Problem number: section 9.3, problem 18.
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]]

\[ \boxed {y^{\prime \prime \prime \prime }-4 y^{\prime \prime \prime }+6 y^{\prime \prime }-4 y^{\prime }+2 y={\mathrm e}^{2 x} \left (x^{4}+x +24\right )} \] 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 \[ y^{\prime \prime \prime \prime }-4 y^{\prime \prime \prime }+6 y^{\prime \prime }-4 y^{\prime }+2 y = 0 \] The characteristic equation is \[ \lambda ^{4}-4 \lambda ^{3}+6 \lambda ^{2}-4 \lambda +2 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}+1\\ \lambda _2 &= -\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}+1\\ \lambda _3 &= -\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}+1\\ \lambda _4 &= \frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}+1 \end {align*}

Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{\left (\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}+1\right ) x} c_{1} +{\mathrm e}^{\left (-\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}+1\right ) x} c_{2} +{\mathrm e}^{\left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}+1\right ) x} c_{3} +{\mathrm e}^{\left (-\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}+1\right ) x} c_{4} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= {\mathrm e}^{\left (\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}+1\right ) x} \\ y_2 &= {\mathrm e}^{\left (-\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}+1\right ) x} \\ y_3 &= {\mathrm e}^{\left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}+1\right ) x} \\ y_4 &= {\mathrm e}^{\left (-\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}+1\right ) x} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\prime \prime \prime \prime }-4 y^{\prime \prime \prime }+6 y^{\prime \prime }-4 y^{\prime }+2 y = {\mathrm e}^{2 x} \left (x^{4}+x +24\right ) \] The particular solution is found using the method of undetermined coefficients. Looking at the RHS of the ode, which is \[ {\mathrm e}^{2 x} \left (x^{4}+x +24\right ) \] Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is \[ [\{x \,{\mathrm e}^{2 x}, x^{2} {\mathrm e}^{2 x}, {\mathrm e}^{2 x} x^{3}, {\mathrm e}^{2 x} x^{4}, {\mathrm e}^{2 x}\}] \] While the set of the basis functions for the homogeneous solution found earlier is \[ \left \{{\mathrm e}^{\left (-\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}+1\right ) x}, {\mathrm e}^{\left (-\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}+1\right ) x}, {\mathrm e}^{\left (\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}+1\right ) x}, {\mathrm e}^{\left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}+1\right ) x}\right \} \] Since there is no duplication between the basis function in the UC_set and the basis functions of the homogeneous solution, the trial solution is a linear combination of all the basis in the UC_set. \[ y_p = A_{1} x \,{\mathrm e}^{2 x}+A_{2} x^{2} {\mathrm e}^{2 x}+A_{3} {\mathrm e}^{2 x} x^{3}+A_{4} {\mathrm e}^{2 x} x^{4}+A_{5} {\mathrm e}^{2 x} \] The unknowns \(\{A_{1}, A_{2}, A_{3}, A_{4}, A_{5}\}\) are found by substituting the above trial solution \(y_p\) into the ODE and comparing coefficients. Substituting the trial solution into the ODE and simplifying gives \[ 36 A_{3} {\mathrm e}^{2 x} x +72 A_{4} {\mathrm e}^{2 x} x^{2}+96 A_{4} {\mathrm e}^{2 x} x +8 A_{2} x \,{\mathrm e}^{2 x}+12 A_{3} {\mathrm e}^{2 x} x^{2}+16 A_{4} {\mathrm e}^{2 x} x^{3}+2 A_{5} {\mathrm e}^{2 x}+4 A_{1} {\mathrm e}^{2 x}+12 A_{2} {\mathrm e}^{2 x}+2 A_{1} x \,{\mathrm e}^{2 x}+2 A_{2} x^{2} {\mathrm e}^{2 x}+2 A_{3} {\mathrm e}^{2 x} x^{3}+2 A_{4} {\mathrm e}^{2 x} x^{4}+24 A_{3} {\mathrm e}^{2 x}+24 A_{4} {\mathrm e}^{2 x} = {\mathrm e}^{2 x} \left (x^{4}+x +24\right ) \] Solving for the unknowns by comparing coefficients results in \[ \left [A_{1} = {\frac {49}{2}}, A_{2} = 6, A_{3} = -4, A_{4} = {\frac {1}{2}}, A_{5} = -31\right ] \] Substituting the above back in the above trial solution \(y_p\), gives the particular solution \[ y_p = \frac {49 x \,{\mathrm e}^{2 x}}{2}+6 x^{2} {\mathrm e}^{2 x}-4 \,{\mathrm e}^{2 x} x^{3}+\frac {{\mathrm e}^{2 x} x^{4}}{2}-31 \,{\mathrm e}^{2 x} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left ({\mathrm e}^{\left (\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}+1\right ) x} c_{1} +{\mathrm e}^{\left (-\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}+1\right ) x} c_{2} +{\mathrm e}^{\left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}+1\right ) x} c_{3} +{\mathrm e}^{\left (-\frac {\sqrt {2}}{2}-\frac {i \sqrt {2}}{2}+1\right ) x} c_{4}\right ) + \left (\frac {49 x \,{\mathrm e}^{2 x}}{2}+6 x^{2} {\mathrm e}^{2 x}-4 \,{\mathrm e}^{2 x} x^{3}+\frac {{\mathrm e}^{2 x} x^{4}}{2}-31 \,{\mathrm e}^{2 x}\right ) \\ \end{align*} Which simplifies to \[ y = {\mathrm e}^{-\frac {x \left (-2+\left (-1+i\right ) \sqrt {2}\right )}{2}} c_{1} +{\mathrm e}^{\frac {x \left (2+\left (-1+i\right ) \sqrt {2}\right )}{2}} c_{2} +{\mathrm e}^{\frac {x \left (2+\left (1+i\right ) \sqrt {2}\right )}{2}} c_{3} +{\mathrm e}^{-\frac {x \left (-2+\left (1+i\right ) \sqrt {2}\right )}{2}} c_{4} +\frac {49 x \,{\mathrm e}^{2 x}}{2}+6 x^{2} {\mathrm e}^{2 x}-4 \,{\mathrm e}^{2 x} x^{3}+\frac {{\mathrm e}^{2 x} x^{4}}{2}-31 \,{\mathrm e}^{2 x} \]

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{-\frac {x \left (-2+\left (-1+i\right ) \sqrt {2}\right )}{2}} c_{1} +{\mathrm e}^{\frac {x \left (2+\left (-1+i\right ) \sqrt {2}\right )}{2}} c_{2} +{\mathrm e}^{\frac {x \left (2+\left (1+i\right ) \sqrt {2}\right )}{2}} c_{3} +{\mathrm e}^{-\frac {x \left (-2+\left (1+i\right ) \sqrt {2}\right )}{2}} c_{4} +\frac {49 x \,{\mathrm e}^{2 x}}{2}+6 x^{2} {\mathrm e}^{2 x}-4 \,{\mathrm e}^{2 x} x^{3}+\frac {{\mathrm e}^{2 x} x^{4}}{2}-31 \,{\mathrm e}^{2 x} \\ \end{align*}

Verification of solutions

\[ y = {\mathrm e}^{-\frac {x \left (-2+\left (-1+i\right ) \sqrt {2}\right )}{2}} c_{1} +{\mathrm e}^{\frac {x \left (2+\left (-1+i\right ) \sqrt {2}\right )}{2}} c_{2} +{\mathrm e}^{\frac {x \left (2+\left (1+i\right ) \sqrt {2}\right )}{2}} c_{3} +{\mathrm e}^{-\frac {x \left (-2+\left (1+i\right ) \sqrt {2}\right )}{2}} c_{4} +\frac {49 x \,{\mathrm e}^{2 x}}{2}+6 x^{2} {\mathrm e}^{2 x}-4 \,{\mathrm e}^{2 x} x^{3}+\frac {{\mathrm e}^{2 x} x^{4}}{2}-31 \,{\mathrm e}^{2 x} \] Verified OK.

19.18.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime \prime \prime \prime }-4 y^{\prime \prime \prime }+6 y^{\prime \prime }-4 y^{\prime }+2 y={\mathrm e}^{2 x} \left (x^{4}+x +24\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 )={\mathrm e}^{2 x} x^{4}+x \,{\mathrm e}^{2 x}+4 y_{4}\left (x \right )-6 y_{3}\left (x \right )+4 y_{2}\left (x \right )-2 y_{1}\left (x \right )+24 \,{\mathrm e}^{2 x} \\ & {} & \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 )={\mathrm e}^{2 x} x^{4}+x \,{\mathrm e}^{2 x}+4 y_{4}\left (x \right )-6 y_{3}\left (x \right )+4 y_{2}\left (x \right )-2 y_{1}\left (x \right )+24 \,{\mathrm e}^{2 x}\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 \\ -2 & 4 & -6 & 4 \end {array}\right ]\cdot {\moverset {\rightarrow }{y}}\left (x \right )+\left [\begin {array}{c} 0 \\ 0 \\ 0 \\ {\mathrm e}^{2 x} x^{4}+x \,{\mathrm e}^{2 x}+24 \,{\mathrm e}^{2 x} \end {array}\right ] \\ \bullet & {} & \textrm {Define the forcing function}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{f}}\left (x \right )=\left [\begin {array}{c} 0 \\ 0 \\ 0 \\ {\mathrm e}^{2 x} x^{4}+x \,{\mathrm e}^{2 x}+24 \,{\mathrm e}^{2 x} \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 \\ -2 & 4 & -6 & 4 \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 [-\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}+1, \left [\begin {array}{c} \frac {1}{\left (-\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}+1\right )^{3}} \\ \frac {1}{\left (-\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}+1\right )^{2}} \\ \frac {1}{-\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}+1} \\ 1 \end {array}\right ]\right ], \left [-\frac {\sqrt {2}}{2}+\frac {\mathrm {I} \sqrt {2}}{2}+1, \left [\begin {array}{c} \frac {1}{\left (-\frac {\sqrt {2}}{2}+\frac {\mathrm {I} \sqrt {2}}{2}+1\right )^{3}} \\ \frac {1}{\left (-\frac {\sqrt {2}}{2}+\frac {\mathrm {I} \sqrt {2}}{2}+1\right )^{2}} \\ \frac {1}{-\frac {\sqrt {2}}{2}+\frac {\mathrm {I} \sqrt {2}}{2}+1} \\ 1 \end {array}\right ]\right ], \left [\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}+1, \left [\begin {array}{c} \frac {1}{\left (\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}+1\right )^{3}} \\ \frac {1}{\left (\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}+1\right )^{2}} \\ \frac {1}{\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}+1} \\ 1 \end {array}\right ]\right ], \left [\frac {\sqrt {2}}{2}+\frac {\mathrm {I} \sqrt {2}}{2}+1, \left [\begin {array}{c} \frac {1}{\left (\frac {\sqrt {2}}{2}+\frac {\mathrm {I} \sqrt {2}}{2}+1\right )^{3}} \\ \frac {1}{\left (\frac {\sqrt {2}}{2}+\frac {\mathrm {I} \sqrt {2}}{2}+1\right )^{2}} \\ \frac {1}{\frac {\sqrt {2}}{2}+\frac {\mathrm {I} \sqrt {2}}{2}+1} \\ 1 \end {array}\right ]\right ]\right ] \\ \bullet & {} & \textrm {Consider complex eigenpair, complex conjugate eigenvalue can be ignored}\hspace {3pt} \\ {} & {} & \left [-\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}+1, \left [\begin {array}{c} \frac {1}{\left (-\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}+1\right )^{3}} \\ \frac {1}{\left (-\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}+1\right )^{2}} \\ \frac {1}{-\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}+1} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution from eigenpair}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (-\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}+1\right ) x}\cdot \left [\begin {array}{c} \frac {1}{\left (-\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}+1\right )^{3}} \\ \frac {1}{\left (-\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}+1\right )^{2}} \\ \frac {1}{-\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}+1} \\ 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}^{\left (-\frac {\sqrt {2}}{2}+1\right ) x}\cdot \left (\cos \left (\frac {\sqrt {2}\, x}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {2}\, x}{2}\right )\right )\cdot \left [\begin {array}{c} \frac {1}{\left (-\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}+1\right )^{3}} \\ \frac {1}{\left (-\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}+1\right )^{2}} \\ \frac {1}{-\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}+1} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Simplify expression}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (-\frac {\sqrt {2}}{2}+1\right ) x}\cdot \left [\begin {array}{c} \frac {\cos \left (\frac {\sqrt {2}\, x}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {2}\, x}{2}\right )}{\left (-\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}+1\right )^{3}} \\ \frac {\cos \left (\frac {\sqrt {2}\, x}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {2}\, x}{2}\right )}{\left (-\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}+1\right )^{2}} \\ \frac {\cos \left (\frac {\sqrt {2}\, x}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {2}\, x}{2}\right )}{-\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}+1} \\ \cos \left (\frac {\sqrt {2}\, x}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {2}\, x}{2}\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}^{\left (-\frac {\sqrt {2}}{2}+1\right ) x}\cdot \left [\begin {array}{c} \frac {\cos \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}-2 \sin \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}-\cos \left (\frac {\sqrt {2}\, x}{2}\right )+3 \sin \left (\frac {\sqrt {2}\, x}{2}\right )}{\left (-2+\sqrt {2}\right )^{3}} \\ -\frac {\cos \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}-\sin \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}-\cos \left (\frac {\sqrt {2}\, x}{2}\right )+\sin \left (\frac {\sqrt {2}\, x}{2}\right )}{\left (-2+\sqrt {2}\right )^{2}} \\ \frac {\cos \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}-\sin \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}-2 \cos \left (\frac {\sqrt {2}\, x}{2}\right )}{2 \left (-2+\sqrt {2}\right )} \\ \cos \left (\frac {\sqrt {2}\, x}{2}\right ) \end {array}\right ], {\moverset {\rightarrow }{y}}_{2}\left (x \right )={\mathrm e}^{\left (-\frac {\sqrt {2}}{2}+1\right ) x}\cdot \left [\begin {array}{c} -\frac {2 \cos \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}+\sin \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}-3 \cos \left (\frac {\sqrt {2}\, x}{2}\right )-\sin \left (\frac {\sqrt {2}\, x}{2}\right )}{\left (-2+\sqrt {2}\right )^{3}} \\ \frac {\cos \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}+\sin \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}-\cos \left (\frac {\sqrt {2}\, x}{2}\right )-\sin \left (\frac {\sqrt {2}\, x}{2}\right )}{\left (-2+\sqrt {2}\right )^{2}} \\ -\frac {\cos \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}+\sin \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}-2 \sin \left (\frac {\sqrt {2}\, x}{2}\right )}{2 \left (-2+\sqrt {2}\right )} \\ -\sin \left (\frac {\sqrt {2}\, x}{2}\right ) \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Consider complex eigenpair, complex conjugate eigenvalue can be ignored}\hspace {3pt} \\ {} & {} & \left [\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}+1, \left [\begin {array}{c} \frac {1}{\left (\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}+1\right )^{3}} \\ \frac {1}{\left (\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}+1\right )^{2}} \\ \frac {1}{\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}+1} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution from eigenpair}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}+1\right ) x}\cdot \left [\begin {array}{c} \frac {1}{\left (\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}+1\right )^{3}} \\ \frac {1}{\left (\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}+1\right )^{2}} \\ \frac {1}{\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}+1} \\ 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}^{\left (\frac {\sqrt {2}}{2}+1\right ) x}\cdot \left (\cos \left (\frac {\sqrt {2}\, x}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {2}\, x}{2}\right )\right )\cdot \left [\begin {array}{c} \frac {1}{\left (\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}+1\right )^{3}} \\ \frac {1}{\left (\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}+1\right )^{2}} \\ \frac {1}{\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}+1} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Simplify expression}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (\frac {\sqrt {2}}{2}+1\right ) x}\cdot \left [\begin {array}{c} \frac {\cos \left (\frac {\sqrt {2}\, x}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {2}\, x}{2}\right )}{\left (\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}+1\right )^{3}} \\ \frac {\cos \left (\frac {\sqrt {2}\, x}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {2}\, x}{2}\right )}{\left (\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}+1\right )^{2}} \\ \frac {\cos \left (\frac {\sqrt {2}\, x}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {2}\, x}{2}\right )}{\frac {\sqrt {2}}{2}-\frac {\mathrm {I} \sqrt {2}}{2}+1} \\ \cos \left (\frac {\sqrt {2}\, x}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {2}\, x}{2}\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}^{\left (\frac {\sqrt {2}}{2}+1\right ) x}\cdot \left [\begin {array}{c} \frac {\cos \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}+2 \sin \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}+\cos \left (\frac {\sqrt {2}\, x}{2}\right )+3 \sin \left (\frac {\sqrt {2}\, x}{2}\right )}{\left (2+\sqrt {2}\right )^{3}} \\ \frac {\cos \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}+\sin \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}+\cos \left (\frac {\sqrt {2}\, x}{2}\right )+\sin \left (\frac {\sqrt {2}\, x}{2}\right )}{\left (2+\sqrt {2}\right )^{2}} \\ \frac {\cos \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}+\sin \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}+2 \cos \left (\frac {\sqrt {2}\, x}{2}\right )}{2 \left (2+\sqrt {2}\right )} \\ \cos \left (\frac {\sqrt {2}\, x}{2}\right ) \end {array}\right ], {\moverset {\rightarrow }{y}}_{4}\left (x \right )={\mathrm e}^{\left (\frac {\sqrt {2}}{2}+1\right ) x}\cdot \left [\begin {array}{c} \frac {2 \cos \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}-\sin \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}+3 \cos \left (\frac {\sqrt {2}\, x}{2}\right )-\sin \left (\frac {\sqrt {2}\, x}{2}\right )}{\left (2+\sqrt {2}\right )^{3}} \\ \frac {\cos \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}-\sin \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}+\cos \left (\frac {\sqrt {2}\, x}{2}\right )-\sin \left (\frac {\sqrt {2}\, x}{2}\right )}{\left (2+\sqrt {2}\right )^{2}} \\ \frac {\cos \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}-\sin \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}-2 \sin \left (\frac {\sqrt {2}\, x}{2}\right )}{2 \left (2+\sqrt {2}\right )} \\ -\sin \left (\frac {\sqrt {2}\, x}{2}\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} \frac {{\mathrm e}^{\left (-\frac {\sqrt {2}}{2}+1\right ) x} \left (\cos \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}-2 \sin \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}-\cos \left (\frac {\sqrt {2}\, x}{2}\right )+3 \sin \left (\frac {\sqrt {2}\, x}{2}\right )\right )}{\left (-2+\sqrt {2}\right )^{3}} & -\frac {{\mathrm e}^{\left (-\frac {\sqrt {2}}{2}+1\right ) x} \left (2 \cos \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}+\sin \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}-3 \cos \left (\frac {\sqrt {2}\, x}{2}\right )-\sin \left (\frac {\sqrt {2}\, x}{2}\right )\right )}{\left (-2+\sqrt {2}\right )^{3}} & \frac {{\mathrm e}^{\left (\frac {\sqrt {2}}{2}+1\right ) x} \left (\cos \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}+2 \sin \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}+\cos \left (\frac {\sqrt {2}\, x}{2}\right )+3 \sin \left (\frac {\sqrt {2}\, x}{2}\right )\right )}{\left (2+\sqrt {2}\right )^{3}} & \frac {{\mathrm e}^{\left (\frac {\sqrt {2}}{2}+1\right ) x} \left (2 \cos \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}-\sin \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}+3 \cos \left (\frac {\sqrt {2}\, x}{2}\right )-\sin \left (\frac {\sqrt {2}\, x}{2}\right )\right )}{\left (2+\sqrt {2}\right )^{3}} \\ -\frac {{\mathrm e}^{\left (-\frac {\sqrt {2}}{2}+1\right ) x} \left (\cos \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}-\sin \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}-\cos \left (\frac {\sqrt {2}\, x}{2}\right )+\sin \left (\frac {\sqrt {2}\, x}{2}\right )\right )}{\left (-2+\sqrt {2}\right )^{2}} & \frac {{\mathrm e}^{\left (-\frac {\sqrt {2}}{2}+1\right ) x} \left (\cos \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}+\sin \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}-\cos \left (\frac {\sqrt {2}\, x}{2}\right )-\sin \left (\frac {\sqrt {2}\, x}{2}\right )\right )}{\left (-2+\sqrt {2}\right )^{2}} & \frac {{\mathrm e}^{\left (\frac {\sqrt {2}}{2}+1\right ) x} \left (\cos \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}+\sin \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}+\cos \left (\frac {\sqrt {2}\, x}{2}\right )+\sin \left (\frac {\sqrt {2}\, x}{2}\right )\right )}{\left (2+\sqrt {2}\right )^{2}} & \frac {{\mathrm e}^{\left (\frac {\sqrt {2}}{2}+1\right ) x} \left (\cos \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}-\sin \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}+\cos \left (\frac {\sqrt {2}\, x}{2}\right )-\sin \left (\frac {\sqrt {2}\, x}{2}\right )\right )}{\left (2+\sqrt {2}\right )^{2}} \\ \frac {{\mathrm e}^{\left (-\frac {\sqrt {2}}{2}+1\right ) x} \left (\cos \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}-\sin \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}-2 \cos \left (\frac {\sqrt {2}\, x}{2}\right )\right )}{2 \left (-2+\sqrt {2}\right )} & -\frac {{\mathrm e}^{\left (-\frac {\sqrt {2}}{2}+1\right ) x} \left (\cos \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}+\sin \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}-2 \sin \left (\frac {\sqrt {2}\, x}{2}\right )\right )}{2 \left (-2+\sqrt {2}\right )} & \frac {{\mathrm e}^{\left (\frac {\sqrt {2}}{2}+1\right ) x} \left (\cos \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}+\sin \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}+2 \cos \left (\frac {\sqrt {2}\, x}{2}\right )\right )}{2 \left (2+\sqrt {2}\right )} & \frac {{\mathrm e}^{\left (\frac {\sqrt {2}}{2}+1\right ) x} \left (\cos \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}-\sin \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}-2 \sin \left (\frac {\sqrt {2}\, x}{2}\right )\right )}{2 \left (2+\sqrt {2}\right )} \\ {\mathrm e}^{\left (-\frac {\sqrt {2}}{2}+1\right ) x} \cos \left (\frac {\sqrt {2}\, x}{2}\right ) & -{\mathrm e}^{\left (-\frac {\sqrt {2}}{2}+1\right ) x} \sin \left (\frac {\sqrt {2}\, x}{2}\right ) & {\mathrm e}^{\left (\frac {\sqrt {2}}{2}+1\right ) x} \cos \left (\frac {\sqrt {2}\, x}{2}\right ) & -{\mathrm e}^{\left (\frac {\sqrt {2}}{2}+1\right ) x} \sin \left (\frac {\sqrt {2}\, x}{2}\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} \frac {{\mathrm e}^{\left (-\frac {\sqrt {2}}{2}+1\right ) x} \left (\cos \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}-2 \sin \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}-\cos \left (\frac {\sqrt {2}\, x}{2}\right )+3 \sin \left (\frac {\sqrt {2}\, x}{2}\right )\right )}{\left (-2+\sqrt {2}\right )^{3}} & -\frac {{\mathrm e}^{\left (-\frac {\sqrt {2}}{2}+1\right ) x} \left (2 \cos \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}+\sin \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}-3 \cos \left (\frac {\sqrt {2}\, x}{2}\right )-\sin \left (\frac {\sqrt {2}\, x}{2}\right )\right )}{\left (-2+\sqrt {2}\right )^{3}} & \frac {{\mathrm e}^{\left (\frac {\sqrt {2}}{2}+1\right ) x} \left (\cos \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}+2 \sin \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}+\cos \left (\frac {\sqrt {2}\, x}{2}\right )+3 \sin \left (\frac {\sqrt {2}\, x}{2}\right )\right )}{\left (2+\sqrt {2}\right )^{3}} & \frac {{\mathrm e}^{\left (\frac {\sqrt {2}}{2}+1\right ) x} \left (2 \cos \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}-\sin \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}+3 \cos \left (\frac {\sqrt {2}\, x}{2}\right )-\sin \left (\frac {\sqrt {2}\, x}{2}\right )\right )}{\left (2+\sqrt {2}\right )^{3}} \\ -\frac {{\mathrm e}^{\left (-\frac {\sqrt {2}}{2}+1\right ) x} \left (\cos \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}-\sin \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}-\cos \left (\frac {\sqrt {2}\, x}{2}\right )+\sin \left (\frac {\sqrt {2}\, x}{2}\right )\right )}{\left (-2+\sqrt {2}\right )^{2}} & \frac {{\mathrm e}^{\left (-\frac {\sqrt {2}}{2}+1\right ) x} \left (\cos \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}+\sin \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}-\cos \left (\frac {\sqrt {2}\, x}{2}\right )-\sin \left (\frac {\sqrt {2}\, x}{2}\right )\right )}{\left (-2+\sqrt {2}\right )^{2}} & \frac {{\mathrm e}^{\left (\frac {\sqrt {2}}{2}+1\right ) x} \left (\cos \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}+\sin \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}+\cos \left (\frac {\sqrt {2}\, x}{2}\right )+\sin \left (\frac {\sqrt {2}\, x}{2}\right )\right )}{\left (2+\sqrt {2}\right )^{2}} & \frac {{\mathrm e}^{\left (\frac {\sqrt {2}}{2}+1\right ) x} \left (\cos \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}-\sin \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}+\cos \left (\frac {\sqrt {2}\, x}{2}\right )-\sin \left (\frac {\sqrt {2}\, x}{2}\right )\right )}{\left (2+\sqrt {2}\right )^{2}} \\ \frac {{\mathrm e}^{\left (-\frac {\sqrt {2}}{2}+1\right ) x} \left (\cos \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}-\sin \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}-2 \cos \left (\frac {\sqrt {2}\, x}{2}\right )\right )}{2 \left (-2+\sqrt {2}\right )} & -\frac {{\mathrm e}^{\left (-\frac {\sqrt {2}}{2}+1\right ) x} \left (\cos \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}+\sin \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}-2 \sin \left (\frac {\sqrt {2}\, x}{2}\right )\right )}{2 \left (-2+\sqrt {2}\right )} & \frac {{\mathrm e}^{\left (\frac {\sqrt {2}}{2}+1\right ) x} \left (\cos \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}+\sin \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}+2 \cos \left (\frac {\sqrt {2}\, x}{2}\right )\right )}{2 \left (2+\sqrt {2}\right )} & \frac {{\mathrm e}^{\left (\frac {\sqrt {2}}{2}+1\right ) x} \left (\cos \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}-\sin \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}-2 \sin \left (\frac {\sqrt {2}\, x}{2}\right )\right )}{2 \left (2+\sqrt {2}\right )} \\ {\mathrm e}^{\left (-\frac {\sqrt {2}}{2}+1\right ) x} \cos \left (\frac {\sqrt {2}\, x}{2}\right ) & -{\mathrm e}^{\left (-\frac {\sqrt {2}}{2}+1\right ) x} \sin \left (\frac {\sqrt {2}\, x}{2}\right ) & {\mathrm e}^{\left (\frac {\sqrt {2}}{2}+1\right ) x} \cos \left (\frac {\sqrt {2}\, x}{2}\right ) & -{\mathrm e}^{\left (\frac {\sqrt {2}}{2}+1\right ) x} \sin \left (\frac {\sqrt {2}\, x}{2}\right ) \end {array}\right ]\cdot \frac {1}{\left [\begin {array}{cccc} \frac {\sqrt {2}-1}{\left (-2+\sqrt {2}\right )^{3}} & -\frac {-3+2 \sqrt {2}}{\left (-2+\sqrt {2}\right )^{3}} & \frac {1+\sqrt {2}}{\left (2+\sqrt {2}\right )^{3}} & \frac {3+2 \sqrt {2}}{\left (2+\sqrt {2}\right )^{3}} \\ -\frac {\sqrt {2}-1}{\left (-2+\sqrt {2}\right )^{2}} & \frac {\sqrt {2}-1}{\left (-2+\sqrt {2}\right )^{2}} & \frac {1+\sqrt {2}}{\left (2+\sqrt {2}\right )^{2}} & \frac {1+\sqrt {2}}{\left (2+\sqrt {2}\right )^{2}} \\ \frac {1}{2} & -\frac {\sqrt {2}}{2 \left (-2+\sqrt {2}\right )} & \frac {1}{2} & \frac {\sqrt {2}}{2 \left (2+\sqrt {2}\right )} \\ 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 {\left (\left (-8 \sqrt {2}+12\right ) \cos \left (\frac {\sqrt {2}\, x}{2}\right )-4 \left (\sqrt {2}-1\right ) \sin \left (\frac {\sqrt {2}\, x}{2}\right )\right ) {\mathrm e}^{-\frac {\left (-2+\sqrt {2}\right ) x}{2}}-8 \left (\cos \left (\frac {\sqrt {2}\, x}{2}\right ) \left (\sqrt {2}-\frac {3}{2}\right )+\frac {5 \sin \left (\frac {\sqrt {2}\, x}{2}\right ) \left (\sqrt {2}-\frac {7}{5}\right )}{2}\right ) {\mathrm e}^{\frac {\left (2+\sqrt {2}\right ) x}{2}}}{\left (-2+\sqrt {2}\right )^{3} \left (2+\sqrt {2}\right )^{3} \left (-3+2 \sqrt {2}\right )} & \frac {\left (\left (-8 \sqrt {2}+8\right ) \cos \left (\frac {\sqrt {2}\, x}{2}\right )+\left (24 \sqrt {2}-32\right ) \sin \left (\frac {\sqrt {2}\, x}{2}\right )\right ) {\mathrm e}^{\frac {\left (2+\sqrt {2}\right ) x}{2}}+8 \left (\left (\sqrt {2}-1\right ) \cos \left (\frac {\sqrt {2}\, x}{2}\right )+\sin \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}\right ) {\mathrm e}^{-\frac {\left (-2+\sqrt {2}\right ) x}{2}}}{\left (-2+\sqrt {2}\right )^{3} \left (-3+2 \sqrt {2}\right ) \left (\sqrt {2}-1\right ) \left (4+3 \sqrt {2}\right ) \left (2+\sqrt {2}\right )^{3}} & \frac {\left (\left (-12 \sqrt {2}+12\right ) \cos \left (\frac {\sqrt {2}\, x}{2}\right )+\left (-8 \sqrt {2}+4\right ) \sin \left (\frac {\sqrt {2}\, x}{2}\right )\right ) {\mathrm e}^{-\frac {\left (-2+\sqrt {2}\right ) x}{2}}+12 \,{\mathrm e}^{\frac {\left (2+\sqrt {2}\right ) x}{2}} \left (\left (\sqrt {2}-1\right ) \cos \left (\frac {\sqrt {2}\, x}{2}\right )-\frac {4 \sin \left (\frac {\sqrt {2}\, x}{2}\right ) \left (\sqrt {2}-\frac {5}{4}\right )}{3}\right )}{\left (-2+\sqrt {2}\right )^{3} \left (-3+2 \sqrt {2}\right ) \left (\sqrt {2}-1\right ) \left (4+3 \sqrt {2}\right ) \left (2+\sqrt {2}\right )^{3}} & \frac {\left (4 \cos \left (\frac {\sqrt {2}\, x}{2}\right )+4 \sin \left (\frac {\sqrt {2}\, x}{2}\right )\right ) {\mathrm e}^{-\frac {\left (-2+\sqrt {2}\right ) x}{2}}-4 \left (\cos \left (\frac {\sqrt {2}\, x}{2}\right )-\sin \left (\frac {\sqrt {2}\, x}{2}\right )\right ) {\mathrm e}^{\frac {\left (2+\sqrt {2}\right ) x}{2}}}{\left (2+\sqrt {2}\right )^{3} \left (4+3 \sqrt {2}\right ) \left (-3+2 \sqrt {2}\right ) \left (-2+\sqrt {2}\right )^{3}} \\ \frac {6 \left (\sqrt {2}-\frac {4}{3}\right ) \left (\left (\cos \left (\frac {\sqrt {2}\, x}{2}\right )+\sin \left (\frac {\sqrt {2}\, x}{2}\right )\right ) {\mathrm e}^{-\frac {\left (-2+\sqrt {2}\right ) x}{2}}-\left (\cos \left (\frac {\sqrt {2}\, x}{2}\right )-\sin \left (\frac {\sqrt {2}\, x}{2}\right )\right ) {\mathrm e}^{\frac {\left (2+\sqrt {2}\right ) x}{2}}\right )}{\left (-2+\sqrt {2}\right )^{2} \left (2+\sqrt {2}\right )^{2} \left (-3+2 \sqrt {2}\right )} & \frac {\left (\left (-6 \sqrt {2}+4\right ) \cos \left (\frac {\sqrt {2}\, x}{2}\right )+\left (-6 \sqrt {2}+8\right ) \sin \left (\frac {\sqrt {2}\, x}{2}\right )\right ) {\mathrm e}^{-\frac {\left (-2+\sqrt {2}\right ) x}{2}}+10 \left (\left (\sqrt {2}-\frac {6}{5}\right ) \cos \left (\frac {\sqrt {2}\, x}{2}\right )-\frac {\sin \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}}{5}\right ) {\mathrm e}^{\frac {\left (2+\sqrt {2}\right ) x}{2}}}{\left (-2+\sqrt {2}\right )^{2} \left (-3+2 \sqrt {2}\right ) \left (\sqrt {2}-1\right ) \left (4+3 \sqrt {2}\right ) \left (2+\sqrt {2}\right )^{2}} & \frac {\left (\left (8 \sqrt {2}-8\right ) \cos \left (\frac {\sqrt {2}\, x}{2}\right )+\left (8 \sqrt {2}-12\right ) \sin \left (\frac {\sqrt {2}\, x}{2}\right )\right ) {\mathrm e}^{-\frac {\left (-2+\sqrt {2}\right ) x}{2}}-8 \,{\mathrm e}^{\frac {\left (2+\sqrt {2}\right ) x}{2}} \left (\left (\sqrt {2}-1\right ) \cos \left (\frac {\sqrt {2}\, x}{2}\right )-\frac {\sin \left (\frac {\sqrt {2}\, x}{2}\right )}{2}\right )}{\left (-2+\sqrt {2}\right )^{2} \left (-3+2 \sqrt {2}\right ) \left (\sqrt {2}-1\right ) \left (4+3 \sqrt {2}\right ) \left (2+\sqrt {2}\right )^{2}} & \frac {\left (\left (-2 \sqrt {2}+2\right ) \cos \left (\frac {\sqrt {2}\, x}{2}\right )+\left (6-4 \sqrt {2}\right ) \sin \left (\frac {\sqrt {2}\, x}{2}\right )\right ) {\mathrm e}^{-\frac {\left (-2+\sqrt {2}\right ) x}{2}}+2 \left (\left (\sqrt {2}-1\right ) \cos \left (\frac {\sqrt {2}\, x}{2}\right )-\sin \left (\frac {\sqrt {2}\, x}{2}\right )\right ) {\mathrm e}^{\frac {\left (2+\sqrt {2}\right ) x}{2}}}{\left (-2+\sqrt {2}\right )^{2} \left (-3+2 \sqrt {2}\right ) \left (\sqrt {2}-1\right ) \left (4+3 \sqrt {2}\right ) \left (2+\sqrt {2}\right )^{2}} \\ \frac {\left (\cos \left (\frac {\sqrt {2}\, x}{2}\right ) \left (-4+3 \sqrt {2}\right )+\sin \left (\frac {\sqrt {2}\, x}{2}\right ) \left (7 \sqrt {2}-10\right )\right ) {\mathrm e}^{-\frac {\left (-2+\sqrt {2}\right ) x}{2}}-3 \left (\cos \left (\frac {\sqrt {2}\, x}{2}\right ) \left (\sqrt {2}-\frac {4}{3}\right )+\frac {\left (-2+\sqrt {2}\right ) \sin \left (\frac {\sqrt {2}\, x}{2}\right )}{3}\right ) {\mathrm e}^{\frac {\left (2+\sqrt {2}\right ) x}{2}}}{-6+4 \sqrt {2}} & \frac {\left (\left (-\sqrt {2}+1\right ) \cos \left (\frac {\sqrt {2}\, x}{2}\right )+\left (-3 \sqrt {2}+5\right ) \sin \left (\frac {\sqrt {2}\, x}{2}\right )\right ) {\mathrm e}^{-\frac {\left (-2+\sqrt {2}\right ) x}{2}}+\left (\left (\sqrt {2}-1\right ) \cos \left (\frac {\sqrt {2}\, x}{2}\right )+\sin \left (\frac {\sqrt {2}\, x}{2}\right ) \left (\sqrt {2}-3\right )\right ) {\mathrm e}^{\frac {\left (2+\sqrt {2}\right ) x}{2}}}{-2+\sqrt {2}} & \frac {\left (\left (9 \sqrt {2}-14\right ) \sin \left (\frac {\sqrt {2}\, x}{2}\right )+\cos \left (\frac {\sqrt {2}\, x}{2}\right ) \left (-4+3 \sqrt {2}\right )\right ) {\mathrm e}^{-\frac {\left (-2+\sqrt {2}\right ) x}{2}}-{\mathrm e}^{\frac {\left (2+\sqrt {2}\right ) x}{2}} \left (\left (\sqrt {2}-6\right ) \sin \left (\frac {\sqrt {2}\, x}{2}\right )+\cos \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}\right )}{-4+2 \sqrt {2}} & \frac {\sin \left (\frac {\sqrt {2}\, x}{2}\right ) \left (\left (3-2 \sqrt {2}\right ) {\mathrm e}^{-\frac {\left (-2+\sqrt {2}\right ) x}{2}}-{\mathrm e}^{\frac {\left (2+\sqrt {2}\right ) x}{2}}\right )}{-2+\sqrt {2}} \\ \frac {2 \sin \left (\frac {\sqrt {2}\, x}{2}\right ) \left ({\mathrm e}^{-\frac {\left (-2+\sqrt {2}\right ) x}{2}} \left (-3+2 \sqrt {2}\right )+{\mathrm e}^{\frac {\left (2+\sqrt {2}\right ) x}{2}}\right )}{-2+\sqrt {2}} & \frac {\left (\left (3 \sqrt {2}-6\right ) \sin \left (\frac {\sqrt {2}\, x}{2}\right )-\cos \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}\right ) {\mathrm e}^{-\frac {\left (-2+\sqrt {2}\right ) x}{2}}}{2}+\frac {\left (\left (3 \sqrt {2}+6\right ) \sin \left (\frac {\sqrt {2}\, x}{2}\right )+\cos \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}\right ) {\mathrm e}^{\frac {\left (2+\sqrt {2}\right ) x}{2}}}{2} & \frac {\left (\left (-5 \sqrt {2}+8\right ) \sin \left (\frac {\sqrt {2}\, x}{2}\right )+\cos \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}\right ) {\mathrm e}^{-\frac {\left (-2+\sqrt {2}\right ) x}{2}}}{2}-\frac {{\mathrm e}^{\frac {\left (2+\sqrt {2}\right ) x}{2}} \left (\left (5 \sqrt {2}+8\right ) \sin \left (\frac {\sqrt {2}\, x}{2}\right )+\cos \left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}\right )}{2} & \frac {\left (\left (-\sqrt {2}+1\right ) \cos \left (\frac {\sqrt {2}\, x}{2}\right )+\sin \left (\frac {\sqrt {2}\, x}{2}\right ) \left (-3+2 \sqrt {2}\right )\right ) {\mathrm e}^{-\frac {\left (-2+\sqrt {2}\right ) x}{2}}}{2}+\frac {\left (\left (1+\sqrt {2}\right ) \cos \left (\frac {\sqrt {2}\, x}{2}\right )+\sin \left (\frac {\sqrt {2}\, x}{2}\right ) \left (3+2 \sqrt {2}\right )\right ) {\mathrm e}^{\frac {\left (2+\sqrt {2}\right ) x}{2}}}{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 (-2267904 \sqrt {2}+3207296\right ) \cos \left (\frac {\sqrt {2}\, x}{2}\right )+\left (-1956096 \sqrt {2}+2766336\right ) \sin \left (\frac {\sqrt {2}\, x}{2}\right )\right ) {\mathrm e}^{-\frac {\left (-2+\sqrt {2}\right ) x}{2}}+\left (\left (371200 \sqrt {2}-524928\right ) \cos \left (\frac {\sqrt {2}\, x}{2}\right )+\left (-120576 \sqrt {2}+170496\right ) \sin \left (\frac {\sqrt {2}\, x}{2}\right )\right ) {\mathrm e}^{\frac {\left (2+\sqrt {2}\right ) x}{2}}-30592 \,{\mathrm e}^{2 x} \left (x^{4}-8 x^{3}+12 x^{2}+49 x -62\right ) \left (\sqrt {2}-\frac {338}{239}\right )}{\left (-2+\sqrt {2}\right )^{8} \left (-3+2 \sqrt {2}\right ) \left (\sqrt {2}-1\right ) \left (4+3 \sqrt {2}\right ) \left (2+\sqrt {2}\right )^{8} \left (70 \sqrt {2}-99\right )} \\ \frac {\left (\left (1244192 \sqrt {2}-1759552\right ) \cos \left (\frac {\sqrt {2}\, x}{2}\right )+\left (2471456 \sqrt {2}-3495168\right ) \sin \left (\frac {\sqrt {2}\, x}{2}\right )\right ) {\mathrm e}^{-\frac {\left (-2+\sqrt {2}\right ) x}{2}}+\left (\left (-96992 \sqrt {2}+137152\right ) \cos \left (\frac {\sqrt {2}\, x}{2}\right )+\left (-113568 \sqrt {2}+160640\right ) \sin \left (\frac {\sqrt {2}\, x}{2}\right )\right ) {\mathrm e}^{\frac {\left (2+\sqrt {2}\right ) x}{2}}+30592 \,{\mathrm e}^{2 x} \left (\sqrt {2}-\frac {338}{239}\right ) \left (x^{4}-6 x^{3}+61 x -\frac {75}{2}\right )}{\left (-2+\sqrt {2}\right )^{7} \left (-3+2 \sqrt {2}\right ) \left (\sqrt {2}-1\right ) \left (4+3 \sqrt {2}\right ) \left (2+\sqrt {2}\right )^{7} \left (70 \sqrt {2}-99\right )} \\ \frac {\left (\left (-548432 \sqrt {2}+775600\right ) \cos \left (\frac {\sqrt {2}\, x}{2}\right )+\left (-7429520 \sqrt {2}[phone]\right ) \sin \left (\frac {\sqrt {2}\, x}{2}\right )\right ) {\mathrm e}^{-\frac {\left (-2+\sqrt {2}\right ) x}{2}}+\left (\left (-75632 \sqrt {2}+106960\right ) \cos \left (\frac {\sqrt {2}\, x}{2}\right )+\left (148400 \sqrt {2}-209872\right ) \sin \left (\frac {\sqrt {2}\, x}{2}\right )\right ) {\mathrm e}^{\frac {\left (2+\sqrt {2}\right ) x}{2}}-89152 \,{\mathrm e}^{2 x} \left (\sqrt {2}-\frac {1970}{1393}\right ) \left (x^{4}-4 x^{3}-9 x^{2}+61 x -7\right )}{\left (2+\sqrt {2}\right )^{5} \left (70 \sqrt {2}-99\right ) \left (-3+2 \sqrt {2}\right ) \left (-2+\sqrt {2}\right )^{6}} \\ \frac {\left (\left (-740720 \sqrt {2}+1047536\right ) \cos \left (\frac {\sqrt {2}\, x}{2}\right )+\left (2242592 \sqrt {2}-3171504\right ) \sin \left (\frac {\sqrt {2}\, x}{2}\right )\right ) {\mathrm e}^{-\frac {\left (-2+\sqrt {2}\right ) x}{2}}+\left (\left (21808 \sqrt {2}-30832\right ) \cos \left (\frac {\sqrt {2}\, x}{2}\right )+\left (1728 \sqrt {2}-2416\right ) \sin \left (\frac {\sqrt {2}\, x}{2}\right )\right ) {\mathrm e}^{\frac {\left (2+\sqrt {2}\right ) x}{2}}+30592 \,{\mathrm e}^{2 x} \left (\sqrt {2}-\frac {338}{239}\right ) \left (x^{4}-2 x^{3}-15 x^{2}+52 x +\frac {47}{2}\right )}{\left (-2+\sqrt {2}\right )^{6} \left (70 \sqrt {2}-99\right ) \left (2+\sqrt {2}\right )^{5}} \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 (-2267904 \sqrt {2}+3207296\right ) \cos \left (\frac {\sqrt {2}\, x}{2}\right )+\left (-1956096 \sqrt {2}+2766336\right ) \sin \left (\frac {\sqrt {2}\, x}{2}\right )\right ) {\mathrm e}^{-\frac {\left (-2+\sqrt {2}\right ) x}{2}}+\left (\left (371200 \sqrt {2}-524928\right ) \cos \left (\frac {\sqrt {2}\, x}{2}\right )+\left (-120576 \sqrt {2}+170496\right ) \sin \left (\frac {\sqrt {2}\, x}{2}\right )\right ) {\mathrm e}^{\frac {\left (2+\sqrt {2}\right ) x}{2}}-30592 \,{\mathrm e}^{2 x} \left (x^{4}-8 x^{3}+12 x^{2}+49 x -62\right ) \left (\sqrt {2}-\frac {338}{239}\right )}{\left (-2+\sqrt {2}\right )^{8} \left (-3+2 \sqrt {2}\right ) \left (\sqrt {2}-1\right ) \left (4+3 \sqrt {2}\right ) \left (2+\sqrt {2}\right )^{8} \left (70 \sqrt {2}-99\right )} \\ \frac {\left (\left (1244192 \sqrt {2}-1759552\right ) \cos \left (\frac {\sqrt {2}\, x}{2}\right )+\left (2471456 \sqrt {2}-3495168\right ) \sin \left (\frac {\sqrt {2}\, x}{2}\right )\right ) {\mathrm e}^{-\frac {\left (-2+\sqrt {2}\right ) x}{2}}+\left (\left (-96992 \sqrt {2}+137152\right ) \cos \left (\frac {\sqrt {2}\, x}{2}\right )+\left (-113568 \sqrt {2}+160640\right ) \sin \left (\frac {\sqrt {2}\, x}{2}\right )\right ) {\mathrm e}^{\frac {\left (2+\sqrt {2}\right ) x}{2}}+30592 \,{\mathrm e}^{2 x} \left (\sqrt {2}-\frac {338}{239}\right ) \left (x^{4}-6 x^{3}+61 x -\frac {75}{2}\right )}{\left (-2+\sqrt {2}\right )^{7} \left (-3+2 \sqrt {2}\right ) \left (\sqrt {2}-1\right ) \left (4+3 \sqrt {2}\right ) \left (2+\sqrt {2}\right )^{7} \left (70 \sqrt {2}-99\right )} \\ \frac {\left (\left (-548432 \sqrt {2}+775600\right ) \cos \left (\frac {\sqrt {2}\, x}{2}\right )+\left (-7429520 \sqrt {2}[phone]\right ) \sin \left (\frac {\sqrt {2}\, x}{2}\right )\right ) {\mathrm e}^{-\frac {\left (-2+\sqrt {2}\right ) x}{2}}+\left (\left (-75632 \sqrt {2}+106960\right ) \cos \left (\frac {\sqrt {2}\, x}{2}\right )+\left (148400 \sqrt {2}-209872\right ) \sin \left (\frac {\sqrt {2}\, x}{2}\right )\right ) {\mathrm e}^{\frac {\left (2+\sqrt {2}\right ) x}{2}}-89152 \,{\mathrm e}^{2 x} \left (\sqrt {2}-\frac {1970}{1393}\right ) \left (x^{4}-4 x^{3}-9 x^{2}+61 x -7\right )}{\left (2+\sqrt {2}\right )^{5} \left (70 \sqrt {2}-99\right ) \left (-3+2 \sqrt {2}\right ) \left (-2+\sqrt {2}\right )^{6}} \\ \frac {\left (\left (-740720 \sqrt {2}+1047536\right ) \cos \left (\frac {\sqrt {2}\, x}{2}\right )+\left (2242592 \sqrt {2}-3171504\right ) \sin \left (\frac {\sqrt {2}\, x}{2}\right )\right ) {\mathrm e}^{-\frac {\left (-2+\sqrt {2}\right ) x}{2}}+\left (\left (21808 \sqrt {2}-30832\right ) \cos \left (\frac {\sqrt {2}\, x}{2}\right )+\left (1728 \sqrt {2}-2416\right ) \sin \left (\frac {\sqrt {2}\, x}{2}\right )\right ) {\mathrm e}^{\frac {\left (2+\sqrt {2}\right ) x}{2}}+30592 \,{\mathrm e}^{2 x} \left (\sqrt {2}-\frac {338}{239}\right ) \left (x^{4}-2 x^{3}-15 x^{2}+52 x +\frac {47}{2}\right )}{\left (-2+\sqrt {2}\right )^{6} \left (70 \sqrt {2}-99\right ) \left (2+\sqrt {2}\right )^{5}} \end {array}\right ] \\ \bullet & {} & \textrm {First component of the vector is the solution to the ODE}\hspace {3pt} \\ {} & {} & y=\frac {\left (\left (\left (-3712 c_{1} +8960 c_{2} -2267904\right ) \sqrt {2}+5248 c_{1} -12672 c_{2} +3207296\right ) \cos \left (\frac {\sqrt {2}\, x}{2}\right )+8960 \left (\left (c_{1} +\frac {29 c_{2}}{70}-\frac {7641}{35}\right ) \sqrt {2}-\frac {99 c_{1}}{70}-\frac {41 c_{2}}{70}+\frac {10806}{35}\right ) \sin \left (\frac {\sqrt {2}\, x}{2}\right )\right ) {\mathrm e}^{-\frac {\left (-2+\sqrt {2}\right ) x}{2}}+\left (\left (\left (126080 c_{3} -52224 c_{4} +371200\right ) \sqrt {2}-178304 c_{3} +73856 c_{4} -524928\right ) \cos \left (\frac {\sqrt {2}\, x}{2}\right )-52224 \left (\left (c_{3} +\frac {985 c_{4}}{408}+\frac {157}{68}\right ) \sqrt {2}-\frac {577 c_{3}}{408}-\frac {1393 c_{4}}{408}-\frac {111}{34}\right ) \sin \left (\frac {\sqrt {2}\, x}{2}\right )\right ) {\mathrm e}^{\frac {\left (2+\sqrt {2}\right ) x}{2}}-30592 \,{\mathrm e}^{2 x} \left (x^{4}-8 x^{3}+12 x^{2}+49 x -62\right ) \left (\sqrt {2}-\frac {338}{239}\right )}{\left (-2+\sqrt {2}\right )^{8} \left (-3+2 \sqrt {2}\right ) \left (\sqrt {2}-1\right ) \left (4+3 \sqrt {2}\right ) \left (2+\sqrt {2}\right )^{8} \left (70 \sqrt {2}-99\right )} \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: 0.016 (sec). Leaf size: 87

dsolve(diff(y(x),x$4)-4*diff(y(x),x$3)+6*diff(y(x),x$2)-4*diff(y(x),x)+2*y(x)=exp(2*x)*(24+x+x^4),y(x), singsol=all)
 

\[ y \left (x \right ) = {\mathrm e}^{\frac {\left (2+\sqrt {2}\right ) x}{2}} \left (\cos \left (\frac {\sqrt {2}\, x}{2}\right ) c_{1} +\sin \left (\frac {\sqrt {2}\, x}{2}\right ) c_{2} \right )+\left (\cos \left (\frac {\sqrt {2}\, x}{2}\right ) c_{3} +\sin \left (\frac {\sqrt {2}\, x}{2}\right ) c_{4} \right ) {\mathrm e}^{-\frac {\left (\sqrt {2}-2\right ) x}{2}}+\frac {{\mathrm e}^{2 x} \left (x^{4}-8 x^{3}+12 x^{2}+49 x -62\right )}{2} \]

✓ Solution by Mathematica

Time used: 0.013 (sec). Leaf size: 102

DSolve[y''''[x]-4*y'''[x]+6*y''[x]-4*y'[x]+2*y[x]==Exp[2*x]*(24+x+x^4),y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{2} e^{x-\frac {x}{\sqrt {2}}} \left (e^{\frac {x}{\sqrt {2}}+x} \left (x^4-8 x^3+12 x^2+49 x-62\right )+2 \left (c_4 e^{\sqrt {2} x}+c_2\right ) \cos \left (\frac {x}{\sqrt {2}}\right )+2 \left (c_1 e^{\sqrt {2} x}+c_3\right ) \sin \left (\frac {x}{\sqrt {2}}\right )\right ) \]