problem 26

Internal problem ID [2165]
Internal ﬁle name [OUTPUT/2165_Monday_February_26_2024_09_18_00_AM_27506519/index.tex]

Book: Diﬀerential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section: Exercise 19, page 86
Problem number: 26.
ODE order: 4.
ODE degree: 1.

The type(s) of ODE detected by this program : "higher_order_linear_constant_coeﬃcients_ODE"

\[ \boxed {y^{\prime \prime \prime \prime }+4 y=5 \sin \left (3 x \right ) {\mathrm e}^{2 x}} \] 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 \[ y^{\prime \prime \prime \prime }+4 y = 0 \] The characteristic equation is \[ \lambda ^{4}+4 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 1-i\\ \lambda _2 &= 1+i\\ \lambda _3 &= -1-i\\ \lambda _4 &= -1+i \end {align*}

Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{\left (-1-i\right ) x} c_{1} +{\mathrm e}^{\left (1+i\right ) x} c_{2} +{\mathrm e}^{\left (1-i\right ) x} c_{3} +{\mathrm e}^{\left (-1+i\right ) x} c_{4} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= {\mathrm e}^{\left (-1-i\right ) x} \\ y_2 &= {\mathrm e}^{\left (1+i\right ) x} \\ y_3 &= {\mathrm e}^{\left (1-i\right ) x} \\ y_4 &= {\mathrm e}^{\left (-1+i\right ) x} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\prime \prime \prime \prime }+4 y = 5 \sin \left (3 x \right ) {\mathrm e}^{2 x} \] The particular solution is found using the method of undetermined coeﬃcients. Looking at the RHS of the ode, which is \[ 5 \sin \left (3 x \right ) {\mathrm e}^{2 x} \] Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is \[ [\{{\mathrm e}^{2 x} \cos \left (3 x \right ), \sin \left (3 x \right ) {\mathrm e}^{2 x}\}] \] While the set of the basis functions for the homogeneous solution found earlier is \[ \{{\mathrm e}^{\left (-1-i\right ) x}, {\mathrm e}^{\left (-1+i\right ) x}, {\mathrm e}^{\left (1-i\right ) x}, {\mathrm e}^{\left (1+i\right ) x}\} \] 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. \[ y_p = A_{1} {\mathrm e}^{2 x} \cos \left (3 x \right )+A_{2} \sin \left (3 x \right ) {\mathrm e}^{2 x} \] The unknowns \(\{A_{1}, A_{2}\}\) 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 \[ -115 A_{1} {\mathrm e}^{2 x} \cos \left (3 x \right )+120 A_{1} {\mathrm e}^{2 x} \sin \left (3 x \right )-115 A_{2} \sin \left (3 x \right ) {\mathrm e}^{2 x}-120 A_{2} \cos \left (3 x \right ) {\mathrm e}^{2 x} = 5 \sin \left (3 x \right ) {\mathrm e}^{2 x} \] Solving for the unknowns by comparing coeﬃcients results in \[ \left [A_{1} = {\frac {24}{1105}}, A_{2} = -{\frac {23}{1105}}\right ] \] Substituting the above back in the above trial solution \(y_p\), gives the particular solution \[ y_p = \frac {24 \,{\mathrm e}^{2 x} \cos \left (3 x \right )}{1105}-\frac {23 \sin \left (3 x \right ) {\mathrm e}^{2 x}}{1105} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left ({\mathrm e}^{\left (-1-i\right ) x} c_{1} +{\mathrm e}^{\left (1+i\right ) x} c_{2} +{\mathrm e}^{\left (1-i\right ) x} c_{3} +{\mathrm e}^{\left (-1+i\right ) x} c_{4}\right ) + \left (\frac {24 \,{\mathrm e}^{2 x} \cos \left (3 x \right )}{1105}-\frac {23 \sin \left (3 x \right ) {\mathrm e}^{2 x}}{1105}\right ) \\ \end{align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{\left (-1-i\right ) x} c_{1} +{\mathrm e}^{\left (1+i\right ) x} c_{2} +{\mathrm e}^{\left (1-i\right ) x} c_{3} +{\mathrm e}^{\left (-1+i\right ) x} c_{4} +\frac {24 \,{\mathrm e}^{2 x} \cos \left (3 x \right )}{1105}-\frac {23 \sin \left (3 x \right ) {\mathrm e}^{2 x}}{1105} \\ \end{align*}

Veriﬁcation of solutions

\[ y = {\mathrm e}^{\left (-1-i\right ) x} c_{1} +{\mathrm e}^{\left (1+i\right ) x} c_{2} +{\mathrm e}^{\left (1-i\right ) x} c_{3} +{\mathrm e}^{\left (-1+i\right ) x} c_{4} +\frac {24 \,{\mathrm e}^{2 x} \cos \left (3 x \right )}{1105}-\frac {23 \sin \left (3 x \right ) {\mathrm e}^{2 x}}{1105} \] Veriﬁed OK.

11.26.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime \prime \prime \prime }+4 y=5 \sin \left (3 x \right ) {\mathrm e}^{2 x} \\ \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 )=5 \sin \left (3 x \right ) {\mathrm e}^{2 x}-4 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 )=5 \sin \left (3 x \right ) {\mathrm e}^{2 x}-4 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 \\ -4 & 0 & 0 & 0 \end {array}\right ]\cdot {\moverset {\rightarrow }{y}}\left (x \right )+\left [\begin {array}{c} 0 \\ 0 \\ 0 \\ 5 \sin \left (3 x \right ) {\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 \\ 5 \sin \left (3 x \right ) {\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 \\ -4 & 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 [-1-\mathrm {I}, \left [\begin {array}{c} \frac {1}{4}+\frac {\mathrm {I}}{4} \\ -\frac {\mathrm {I}}{2} \\ -\frac {1}{2}+\frac {\mathrm {I}}{2} \\ 1 \end {array}\right ]\right ], \left [-1+\mathrm {I}, \left [\begin {array}{c} \frac {1}{4}-\frac {\mathrm {I}}{4} \\ \frac {\mathrm {I}}{2} \\ -\frac {1}{2}-\frac {\mathrm {I}}{2} \\ 1 \end {array}\right ]\right ], \left [1-\mathrm {I}, \left [\begin {array}{c} -\frac {1}{4}+\frac {\mathrm {I}}{4} \\ \frac {\mathrm {I}}{2} \\ \frac {1}{2}+\frac {\mathrm {I}}{2} \\ 1 \end {array}\right ]\right ], \left [1+\mathrm {I}, \left [\begin {array}{c} -\frac {1}{4}-\frac {\mathrm {I}}{4} \\ -\frac {\mathrm {I}}{2} \\ \frac {1}{2}-\frac {\mathrm {I}}{2} \\ 1 \end {array}\right ]\right ]\right ] \\ \bullet & {} & \textrm {Consider complex eigenpair, complex conjugate eigenvalue can be ignored}\hspace {3pt} \\ {} & {} & \left [-1-\mathrm {I}, \left [\begin {array}{c} \frac {1}{4}+\frac {\mathrm {I}}{4} \\ -\frac {\mathrm {I}}{2} \\ -\frac {1}{2}+\frac {\mathrm {I}}{2} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution from eigenpair}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (-1-\mathrm {I}\right ) x}\cdot \left [\begin {array}{c} \frac {1}{4}+\frac {\mathrm {I}}{4} \\ -\frac {\mathrm {I}}{2} \\ -\frac {1}{2}+\frac {\mathrm {I}}{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}^{-x}\cdot \left (\cos \left (x \right )-\mathrm {I} \sin \left (x \right )\right )\cdot \left [\begin {array}{c} \frac {1}{4}+\frac {\mathrm {I}}{4} \\ -\frac {\mathrm {I}}{2} \\ -\frac {1}{2}+\frac {\mathrm {I}}{2} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Simplify expression}\hspace {3pt} \\ {} & {} & {\mathrm e}^{-x}\cdot \left [\begin {array}{c} \left (\frac {1}{4}+\frac {\mathrm {I}}{4}\right ) \left (\cos \left (x \right )-\mathrm {I} \sin \left (x \right )\right ) \\ -\frac {\mathrm {I}}{2} \left (\cos \left (x \right )-\mathrm {I} \sin \left (x \right )\right ) \\ \left (-\frac {1}{2}+\frac {\mathrm {I}}{2}\right ) \left (\cos \left (x \right )-\mathrm {I} \sin \left (x \right )\right ) \\ \cos \left (x \right )-\mathrm {I} \sin \left (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}^{-x}\cdot \left [\begin {array}{c} \frac {\cos \left (x \right )}{4}+\frac {\sin \left (x \right )}{4} \\ -\frac {\sin \left (x \right )}{2} \\ -\frac {\cos \left (x \right )}{2}+\frac {\sin \left (x \right )}{2} \\ \cos \left (x \right ) \end {array}\right ], {\moverset {\rightarrow }{y}}_{2}\left (x \right )={\mathrm e}^{-x}\cdot \left [\begin {array}{c} \frac {\cos \left (x \right )}{4}-\frac {\sin \left (x \right )}{4} \\ -\frac {\cos \left (x \right )}{2} \\ \frac {\cos \left (x \right )}{2}+\frac {\sin \left (x \right )}{2} \\ -\sin \left (x \right ) \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Consider complex eigenpair, complex conjugate eigenvalue can be ignored}\hspace {3pt} \\ {} & {} & \left [1-\mathrm {I}, \left [\begin {array}{c} -\frac {1}{4}+\frac {\mathrm {I}}{4} \\ \frac {\mathrm {I}}{2} \\ \frac {1}{2}+\frac {\mathrm {I}}{2} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution from eigenpair}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (1-\mathrm {I}\right ) x}\cdot \left [\begin {array}{c} -\frac {1}{4}+\frac {\mathrm {I}}{4} \\ \frac {\mathrm {I}}{2} \\ \frac {1}{2}+\frac {\mathrm {I}}{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}^{x}\cdot \left (\cos \left (x \right )-\mathrm {I} \sin \left (x \right )\right )\cdot \left [\begin {array}{c} -\frac {1}{4}+\frac {\mathrm {I}}{4} \\ \frac {\mathrm {I}}{2} \\ \frac {1}{2}+\frac {\mathrm {I}}{2} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Simplify expression}\hspace {3pt} \\ {} & {} & {\mathrm e}^{x}\cdot \left [\begin {array}{c} \left (-\frac {1}{4}+\frac {\mathrm {I}}{4}\right ) \left (\cos \left (x \right )-\mathrm {I} \sin \left (x \right )\right ) \\ \frac {\mathrm {I}}{2} \left (\cos \left (x \right )-\mathrm {I} \sin \left (x \right )\right ) \\ \left (\frac {1}{2}+\frac {\mathrm {I}}{2}\right ) \left (\cos \left (x \right )-\mathrm {I} \sin \left (x \right )\right ) \\ \cos \left (x \right )-\mathrm {I} \sin \left (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}^{x}\cdot \left [\begin {array}{c} -\frac {\cos \left (x \right )}{4}+\frac {\sin \left (x \right )}{4} \\ \frac {\sin \left (x \right )}{2} \\ \frac {\cos \left (x \right )}{2}+\frac {\sin \left (x \right )}{2} \\ \cos \left (x \right ) \end {array}\right ], {\moverset {\rightarrow }{y}}_{4}\left (x \right )={\mathrm e}^{x}\cdot \left [\begin {array}{c} \frac {\cos \left (x \right )}{4}+\frac {\sin \left (x \right )}{4} \\ \frac {\cos \left (x \right )}{2} \\ \frac {\cos \left (x \right )}{2}-\frac {\sin \left (x \right )}{2} \\ -\sin \left (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}^{-x} \left (\frac {\cos \left (x \right )}{4}+\frac {\sin \left (x \right )}{4}\right ) & {\mathrm e}^{-x} \left (\frac {\cos \left (x \right )}{4}-\frac {\sin \left (x \right )}{4}\right ) & {\mathrm e}^{x} \left (-\frac {\cos \left (x \right )}{4}+\frac {\sin \left (x \right )}{4}\right ) & {\mathrm e}^{x} \left (\frac {\cos \left (x \right )}{4}+\frac {\sin \left (x \right )}{4}\right ) \\ -\frac {\sin \left (x \right ) {\mathrm e}^{-x}}{2} & -\frac {\cos \left (x \right ) {\mathrm e}^{-x}}{2} & \frac {\sin \left (x \right ) {\mathrm e}^{x}}{2} & \frac {\cos \left (x \right ) {\mathrm e}^{x}}{2} \\ {\mathrm e}^{-x} \left (-\frac {\cos \left (x \right )}{2}+\frac {\sin \left (x \right )}{2}\right ) & {\mathrm e}^{-x} \left (\frac {\cos \left (x \right )}{2}+\frac {\sin \left (x \right )}{2}\right ) & {\mathrm e}^{x} \left (\frac {\cos \left (x \right )}{2}+\frac {\sin \left (x \right )}{2}\right ) & \left (\frac {\cos \left (x \right )}{2}-\frac {\sin \left (x \right )}{2}\right ) {\mathrm e}^{x} \\ \cos \left (x \right ) {\mathrm e}^{-x} & -\sin \left (x \right ) {\mathrm e}^{-x} & \cos \left (x \right ) {\mathrm e}^{x} & -\sin \left (x \right ) {\mathrm e}^{x} \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}^{-x} \left (\frac {\cos \left (x \right )}{4}+\frac {\sin \left (x \right )}{4}\right ) & {\mathrm e}^{-x} \left (\frac {\cos \left (x \right )}{4}-\frac {\sin \left (x \right )}{4}\right ) & {\mathrm e}^{x} \left (-\frac {\cos \left (x \right )}{4}+\frac {\sin \left (x \right )}{4}\right ) & {\mathrm e}^{x} \left (\frac {\cos \left (x \right )}{4}+\frac {\sin \left (x \right )}{4}\right ) \\ -\frac {\sin \left (x \right ) {\mathrm e}^{-x}}{2} & -\frac {\cos \left (x \right ) {\mathrm e}^{-x}}{2} & \frac {\sin \left (x \right ) {\mathrm e}^{x}}{2} & \frac {\cos \left (x \right ) {\mathrm e}^{x}}{2} \\ {\mathrm e}^{-x} \left (-\frac {\cos \left (x \right )}{2}+\frac {\sin \left (x \right )}{2}\right ) & {\mathrm e}^{-x} \left (\frac {\cos \left (x \right )}{2}+\frac {\sin \left (x \right )}{2}\right ) & {\mathrm e}^{x} \left (\frac {\cos \left (x \right )}{2}+\frac {\sin \left (x \right )}{2}\right ) & \left (\frac {\cos \left (x \right )}{2}-\frac {\sin \left (x \right )}{2}\right ) {\mathrm e}^{x} \\ \cos \left (x \right ) {\mathrm e}^{-x} & -\sin \left (x \right ) {\mathrm e}^{-x} & \cos \left (x \right ) {\mathrm e}^{x} & -\sin \left (x \right ) {\mathrm e}^{x} \end {array}\right ]\cdot \frac {1}{\left [\begin {array}{cccc} \frac {1}{4} & \frac {1}{4} & -\frac {1}{4} & \frac {1}{4} \\ 0 & -\frac {1}{2} & 0 & \frac {1}{2} \\ -\frac {1}{2} & \frac {1}{2} & \frac {1}{2} & \frac {1}{2} \\ 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 (x \right ) \left ({\mathrm e}^{x}+{\mathrm e}^{-x}\right )}{2} & \frac {{\mathrm e}^{-x} \left (-\cos \left (x \right )+\sin \left (x \right )\right )}{4}+\frac {{\mathrm e}^{x} \left (\cos \left (x \right )+\sin \left (x \right )\right )}{4} & \frac {\sin \left (x \right ) \left ({\mathrm e}^{x}-{\mathrm e}^{-x}\right )}{4} & \frac {{\mathrm e}^{-x} \left (\cos \left (x \right )+\sin \left (x \right )\right )}{8}-\frac {{\mathrm e}^{x} \left (\cos \left (x \right )-\sin \left (x \right )\right )}{8} \\ \frac {\left (-\cos \left (x \right )-\sin \left (x \right )\right ) {\mathrm e}^{-x}}{2}+\frac {{\mathrm e}^{x} \left (\cos \left (x \right )-\sin \left (x \right )\right )}{2} & \frac {\cos \left (x \right ) \left ({\mathrm e}^{x}+{\mathrm e}^{-x}\right )}{2} & \frac {{\mathrm e}^{-x} \left (-\cos \left (x \right )+\sin \left (x \right )\right )}{4}+\frac {{\mathrm e}^{x} \left (\cos \left (x \right )+\sin \left (x \right )\right )}{4} & \frac {\sin \left (x \right ) \left ({\mathrm e}^{x}-{\mathrm e}^{-x}\right )}{4} \\ -\sin \left (x \right ) \left ({\mathrm e}^{x}-{\mathrm e}^{-x}\right ) & \frac {\left (-\cos \left (x \right )-\sin \left (x \right )\right ) {\mathrm e}^{-x}}{2}+\frac {{\mathrm e}^{x} \left (\cos \left (x \right )-\sin \left (x \right )\right )}{2} & \frac {\cos \left (x \right ) \left ({\mathrm e}^{x}+{\mathrm e}^{-x}\right )}{2} & \frac {{\mathrm e}^{-x} \left (-\cos \left (x \right )+\sin \left (x \right )\right )}{4}+\frac {{\mathrm e}^{x} \left (\cos \left (x \right )+\sin \left (x \right )\right )}{4} \\ {\mathrm e}^{-x} \left (\cos \left (x \right )-\sin \left (x \right )\right )-{\mathrm e}^{x} \left (\cos \left (x \right )+\sin \left (x \right )\right ) & -\sin \left (x \right ) \left ({\mathrm e}^{x}-{\mathrm e}^{-x}\right ) & \frac {\left (-\cos \left (x \right )-\sin \left (x \right )\right ) {\mathrm e}^{-x}}{2}+\frac {{\mathrm e}^{x} \left (\cos \left (x \right )-\sin \left (x \right )\right )}{2} & \frac {\cos \left (x \right ) \left ({\mathrm e}^{x}+{\mathrm e}^{-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 (96 \cos \left (x \right )^{3}-92 \sin \left (x \right ) \cos \left (x \right )^{2}-72 \cos \left (x \right )+23 \sin \left (x \right )\right ) {\mathrm e}^{2 x}}{1105}+\frac {3 \left (11 \sin \left (x \right )+23 \cos \left (x \right )\right ) {\mathrm e}^{-x}}{520}-\frac {21 \,{\mathrm e}^{x} \left (\cos \left (x \right )-\frac {11 \sin \left (x \right )}{7}\right )}{136} \\ \frac {\left (-84 \cos \left (x \right )^{3}-472 \sin \left (x \right ) \cos \left (x \right )^{2}+63 \cos \left (x \right )+118 \sin \left (x \right )\right ) {\mathrm e}^{2 x}}{1105}+\frac {3 \left (-17 \sin \left (x \right )-6 \cos \left (x \right )\right ) {\mathrm e}^{-x}}{260}+\frac {3 \,{\mathrm e}^{x} \left (\frac {9 \sin \left (x \right )}{2}+\cos \left (x \right )\right )}{34} \\ \frac {\left (-1584 \cos \left (x \right )^{3}-692 \sin \left (x \right ) \cos \left (x \right )^{2}+1188 \cos \left (x \right )+173 \sin \left (x \right )\right ) {\mathrm e}^{2 x}}{1105}+\frac {3 \left (23 \sin \left (x \right )-11 \cos \left (x \right )\right ) {\mathrm e}^{-x}}{260}+\frac {33 \,{\mathrm e}^{x} \left (\cos \left (x \right )+\frac {7 \sin \left (x \right )}{11}\right )}{68} \\ \frac {\left (-5244 \cos \left (x \right )^{3}+3368 \sin \left (x \right ) \cos \left (x \right )^{2}+3933 \cos \left (x \right )-842 \sin \left (x \right )\right ) {\mathrm e}^{2 x}}{1105}+\frac {3 \left (-6 \sin \left (x \right )+17 \cos \left (x \right )\right ) {\mathrm e}^{-x}}{130}+\frac {27 \,{\mathrm e}^{x} \left (\cos \left (x \right )-\frac {2 \sin \left (x \right )}{9}\right )}{34} \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 (96 \cos \left (x \right )^{3}-92 \sin \left (x \right ) \cos \left (x \right )^{2}-72 \cos \left (x \right )+23 \sin \left (x \right )\right ) {\mathrm e}^{2 x}}{1105}+\frac {3 \left (11 \sin \left (x \right )+23 \cos \left (x \right )\right ) {\mathrm e}^{-x}}{520}-\frac {21 \,{\mathrm e}^{x} \left (\cos \left (x \right )-\frac {11 \sin \left (x \right )}{7}\right )}{136} \\ \frac {\left (-84 \cos \left (x \right )^{3}-472 \sin \left (x \right ) \cos \left (x \right )^{2}+63 \cos \left (x \right )+118 \sin \left (x \right )\right ) {\mathrm e}^{2 x}}{1105}+\frac {3 \left (-17 \sin \left (x \right )-6 \cos \left (x \right )\right ) {\mathrm e}^{-x}}{260}+\frac {3 \,{\mathrm e}^{x} \left (\frac {9 \sin \left (x \right )}{2}+\cos \left (x \right )\right )}{34} \\ \frac {\left (-1584 \cos \left (x \right )^{3}-692 \sin \left (x \right ) \cos \left (x \right )^{2}+1188 \cos \left (x \right )+173 \sin \left (x \right )\right ) {\mathrm e}^{2 x}}{1105}+\frac {3 \left (23 \sin \left (x \right )-11 \cos \left (x \right )\right ) {\mathrm e}^{-x}}{260}+\frac {33 \,{\mathrm e}^{x} \left (\cos \left (x \right )+\frac {7 \sin \left (x \right )}{11}\right )}{68} \\ \frac {\left (-5244 \cos \left (x \right )^{3}+3368 \sin \left (x \right ) \cos \left (x \right )^{2}+3933 \cos \left (x \right )-842 \sin \left (x \right )\right ) {\mathrm e}^{2 x}}{1105}+\frac {3 \left (-6 \sin \left (x \right )+17 \cos \left (x \right )\right ) {\mathrm e}^{-x}}{130}+\frac {27 \,{\mathrm e}^{x} \left (\cos \left (x \right )-\frac {2 \sin \left (x \right )}{9}\right )}{34} \end {array}\right ] \\ \bullet & {} & \textrm {First component of the vector is the solution to the ODE}\hspace {3pt} \\ {} & {} & y=\frac {\left (\left (69+130 c_{1} +130 c_{2} \right ) \cos \left (x \right )+130 \sin \left (x \right ) \left (c_{1} -c_{2} +\frac {33}{130}\right )\right ) {\mathrm e}^{-x}}{520}+\frac {\left (96 \cos \left (x \right )^{3}-92 \sin \left (x \right ) \cos \left (x \right )^{2}-72 \cos \left (x \right )+23 \sin \left (x \right )\right ) {\mathrm e}^{2 x}}{1105}-\frac {\left (\left (c_{3} -c_{4} +\frac {21}{34}\right ) \cos \left (x \right )-\sin \left (x \right ) \left (c_{3} +c_{4} +\frac {33}{34}\right )\right ) {\mathrm e}^{x}}{4} \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`

\[ y \left (x \right ) = \left (\cos \left (x \right ) c_{3} +c_{4} \sin \left (x \right )\right ) {\mathrm e}^{-x}+\frac {{\mathrm e}^{2 x} \left (24 \cos \left (3 x \right )-23 \sin \left (3 x \right )\right )}{1105}+{\mathrm e}^{x} \left (\cos \left (x \right ) c_{1} +\sin \left (x \right ) c_{2} \right ) \]

\[ y(x)\to \frac {e^{2 x} (24 \cos (3 x)-23 \sin (3 x))}{1105}+c_1 e^{-x} \cos (x)+c_4 e^x \cos (x)+c_2 e^{-x} \sin (x)+c_3 e^x \sin (x) \]

11.26 problem 26

11.26.1 Maple step by step solution