problem section 9.3, problem 39

Internal problem ID [1536]
Internal ﬁle name [OUTPUT/1537_Sunday_June_05_2022_02_21_12_AM_48519893/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 39.
ODE order: 4.
ODE degree: 1.

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

\[ \boxed {y^{\prime \prime \prime \prime }-8 y^{\prime \prime \prime }+24 y^{\prime \prime }-32 y^{\prime }+15 y={\mathrm e}^{2 x} \left (15 x \cos \left (2 x \right )+32 \sin \left (2 x \right )\right )} \] 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 \[ y^{\prime \prime \prime \prime }-8 y^{\prime \prime \prime }+24 y^{\prime \prime }-32 y^{\prime }+15 y = 0 \] The characteristic equation is \[ \lambda ^{4}-8 \lambda ^{3}+24 \lambda ^{2}-32 \lambda +15 = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 3\\ \lambda _2 &= 1\\ \lambda _3 &= 2+i\\ \lambda _4 &= 2-i \end {align*}

Therefore the homogeneous solution is \[ y_h(x)=c_{1} {\mathrm e}^{x}+c_{2} {\mathrm e}^{3 x}+{\mathrm e}^{\left (2-i\right ) x} c_{3} +{\mathrm e}^{\left (2+i\right ) x} c_{4} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= {\mathrm e}^{x} \\ y_2 &= {\mathrm e}^{3 x} \\ y_3 &= {\mathrm e}^{\left (2-i\right ) x} \\ y_4 &= {\mathrm e}^{\left (2+i\right ) x} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\prime \prime \prime \prime }-8 y^{\prime \prime \prime }+24 y^{\prime \prime }-32 y^{\prime }+15 y = {\mathrm e}^{2 x} \left (15 x \cos \left (2 x \right )+32 \sin \left (2 x \right )\right ) \] The particular solution is found using the method of undetermined coeﬃcients. Looking at the RHS of the ode, which is \[ {\mathrm e}^{2 x} \left (15 x \cos \left (2 x \right )+32 \sin \left (2 x \right )\right ) \] Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is \[ [\{{\mathrm e}^{2 x} \cos \left (2 x \right ), {\mathrm e}^{2 x} \sin \left (2 x \right ), {\mathrm e}^{2 x} \cos \left (2 x \right ) x, {\mathrm e}^{2 x} \sin \left (2 x \right ) x\}] \] While the set of the basis functions for the homogeneous solution found earlier is \[ \{{\mathrm e}^{x}, {\mathrm e}^{\left (2-i\right ) x}, {\mathrm e}^{\left (2+i\right ) x}, {\mathrm e}^{3 x}\} \] 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. \[ y_p = A_{1} {\mathrm e}^{2 x} \cos \left (2 x \right )+A_{2} {\mathrm e}^{2 x} \sin \left (2 x \right )+A_{3} {\mathrm e}^{2 x} \cos \left (2 x \right ) x +A_{4} {\mathrm e}^{2 x} \sin \left (2 x \right ) x \] The unknowns \(\{A_{1}, A_{2}, A_{3}, A_{4}\}\) 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 \[ -32 A_{4} {\mathrm e}^{2 x} \cos \left (2 x \right )+32 A_{3} {\mathrm e}^{2 x} \sin \left (2 x \right )+15 A_{4} {\mathrm e}^{2 x} \sin \left (2 x \right ) x +15 A_{3} {\mathrm e}^{2 x} \cos \left (2 x \right ) x +15 A_{1} {\mathrm e}^{2 x} \cos \left (2 x \right )+15 A_{2} {\mathrm e}^{2 x} \sin \left (2 x \right ) = {\mathrm e}^{2 x} \left (15 x \cos \left (2 x \right )+32 \sin \left (2 x \right )\right ) \] Solving for the unknowns by comparing coeﬃcients results in \[ [A_{1} = 0, A_{2} = 0, A_{3} = 1, A_{4} = 0] \] Substituting the above back in the above trial solution \(y_p\), gives the particular solution \[ y_p = {\mathrm e}^{2 x} \cos \left (2 x \right ) x \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left (c_{1} {\mathrm e}^{x}+c_{2} {\mathrm e}^{3 x}+{\mathrm e}^{\left (2-i\right ) x} c_{3} +{\mathrm e}^{\left (2+i\right ) x} c_{4}\right ) + \left ({\mathrm e}^{2 x} \cos \left (2 x \right ) x\right ) \\ \end{align*}

Summary

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

Veriﬁcation of solutions

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

19.39.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime \prime \prime \prime }-8 y^{\prime \prime \prime }+24 y^{\prime \prime }-32 y^{\prime }+15 y={\mathrm e}^{2 x} \left (15 x \cos \left (2 x \right )+32 \sin \left (2 x \right )\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 )=15 \,{\mathrm e}^{2 x} \cos \left (2 x \right ) x +32 \,{\mathrm e}^{2 x} \sin \left (2 x \right )+8 y_{4}\left (x \right )-24 y_{3}\left (x \right )+32 y_{2}\left (x \right )-15 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 )=15 \,{\mathrm e}^{2 x} \cos \left (2 x \right ) x +32 \,{\mathrm e}^{2 x} \sin \left (2 x \right )+8 y_{4}\left (x \right )-24 y_{3}\left (x \right )+32 y_{2}\left (x \right )-15 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 \\ -15 & 32 & -24 & 8 \end {array}\right ]\cdot {\moverset {\rightarrow }{y}}\left (x \right )+\left [\begin {array}{c} 0 \\ 0 \\ 0 \\ 15 \,{\mathrm e}^{2 x} \cos \left (2 x \right ) x +32 \,{\mathrm e}^{2 x} \sin \left (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 \\ 15 \,{\mathrm e}^{2 x} \cos \left (2 x \right ) x +32 \,{\mathrm e}^{2 x} \sin \left (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 \\ -15 & 32 & -24 & 8 \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, \left [\begin {array}{c} 1 \\ 1 \\ 1 \\ 1 \end {array}\right ]\right ], \left [3, \left [\begin {array}{c} \frac {1}{27} \\ \frac {1}{9} \\ \frac {1}{3} \\ 1 \end {array}\right ]\right ], \left [2-\mathrm {I}, \left [\begin {array}{c} \frac {2}{125}+\frac {11 \,\mathrm {I}}{125} \\ \frac {3}{25}+\frac {4 \,\mathrm {I}}{25} \\ \frac {2}{5}+\frac {\mathrm {I}}{5} \\ 1 \end {array}\right ]\right ], \left [2+\mathrm {I}, \left [\begin {array}{c} \frac {2}{125}-\frac {11 \,\mathrm {I}}{125} \\ \frac {3}{25}-\frac {4 \,\mathrm {I}}{25} \\ \frac {2}{5}-\frac {\mathrm {I}}{5} \\ 1 \end {array}\right ]\right ]\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [1, \left [\begin {array}{c} 1 \\ 1 \\ 1 \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{1}={\mathrm e}^{x}\cdot \left [\begin {array}{c} 1 \\ 1 \\ 1 \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [3, \left [\begin {array}{c} \frac {1}{27} \\ \frac {1}{9} \\ \frac {1}{3} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution to homogeneous system from eigenpair}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}_{2}={\mathrm e}^{3 x}\cdot \left [\begin {array}{c} \frac {1}{27} \\ \frac {1}{9} \\ \frac {1}{3} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Consider complex eigenpair, complex conjugate eigenvalue can be ignored}\hspace {3pt} \\ {} & {} & \left [2-\mathrm {I}, \left [\begin {array}{c} \frac {2}{125}+\frac {11 \,\mathrm {I}}{125} \\ \frac {3}{25}+\frac {4 \,\mathrm {I}}{25} \\ \frac {2}{5}+\frac {\mathrm {I}}{5} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution from eigenpair}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (2-\mathrm {I}\right ) x}\cdot \left [\begin {array}{c} \frac {2}{125}+\frac {11 \,\mathrm {I}}{125} \\ \frac {3}{25}+\frac {4 \,\mathrm {I}}{25} \\ \frac {2}{5}+\frac {\mathrm {I}}{5} \\ 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}^{2 x}\cdot \left (\cos \left (x \right )-\mathrm {I} \sin \left (x \right )\right )\cdot \left [\begin {array}{c} \frac {2}{125}+\frac {11 \,\mathrm {I}}{125} \\ \frac {3}{25}+\frac {4 \,\mathrm {I}}{25} \\ \frac {2}{5}+\frac {\mathrm {I}}{5} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Simplify expression}\hspace {3pt} \\ {} & {} & {\mathrm e}^{2 x}\cdot \left [\begin {array}{c} \left (\frac {2}{125}+\frac {11 \,\mathrm {I}}{125}\right ) \left (\cos \left (x \right )-\mathrm {I} \sin \left (x \right )\right ) \\ \left (\frac {3}{25}+\frac {4 \,\mathrm {I}}{25}\right ) \left (\cos \left (x \right )-\mathrm {I} \sin \left (x \right )\right ) \\ \left (\frac {2}{5}+\frac {\mathrm {I}}{5}\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}^{2 x}\cdot \left [\begin {array}{c} \frac {2 \cos \left (x \right )}{125}+\frac {11 \sin \left (x \right )}{125} \\ \frac {3 \cos \left (x \right )}{25}+\frac {4 \sin \left (x \right )}{25} \\ \frac {\sin \left (x \right )}{5}+\frac {2 \cos \left (x \right )}{5} \\ \cos \left (x \right ) \end {array}\right ], {\moverset {\rightarrow }{y}}_{4}\left (x \right )={\mathrm e}^{2 x}\cdot \left [\begin {array}{c} -\frac {2 \sin \left (x \right )}{125}+\frac {11 \cos \left (x \right )}{125} \\ -\frac {3 \sin \left (x \right )}{25}+\frac {4 \cos \left (x \right )}{25} \\ -\frac {2 \sin \left (x \right )}{5}+\frac {\cos \left (x \right )}{5} \\ -\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}+c_{2} {\moverset {\rightarrow }{y}}_{2}+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} & \frac {{\mathrm e}^{3 x}}{27} & {\mathrm e}^{2 x} \left (\frac {2 \cos \left (x \right )}{125}+\frac {11 \sin \left (x \right )}{125}\right ) & {\mathrm e}^{2 x} \left (-\frac {2 \sin \left (x \right )}{125}+\frac {11 \cos \left (x \right )}{125}\right ) \\ {\mathrm e}^{x} & \frac {{\mathrm e}^{3 x}}{9} & {\mathrm e}^{2 x} \left (\frac {3 \cos \left (x \right )}{25}+\frac {4 \sin \left (x \right )}{25}\right ) & {\mathrm e}^{2 x} \left (-\frac {3 \sin \left (x \right )}{25}+\frac {4 \cos \left (x \right )}{25}\right ) \\ {\mathrm e}^{x} & \frac {{\mathrm e}^{3 x}}{3} & {\mathrm e}^{2 x} \left (\frac {\sin \left (x \right )}{5}+\frac {2 \cos \left (x \right )}{5}\right ) & {\mathrm e}^{2 x} \left (-\frac {2 \sin \left (x \right )}{5}+\frac {\cos \left (x \right )}{5}\right ) \\ {\mathrm e}^{x} & {\mathrm e}^{3 x} & {\mathrm e}^{2 x} \cos \left (x \right ) & -{\mathrm e}^{2 x} \sin \left (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}^{x} & \frac {{\mathrm e}^{3 x}}{27} & {\mathrm e}^{2 x} \left (\frac {2 \cos \left (x \right )}{125}+\frac {11 \sin \left (x \right )}{125}\right ) & {\mathrm e}^{2 x} \left (-\frac {2 \sin \left (x \right )}{125}+\frac {11 \cos \left (x \right )}{125}\right ) \\ {\mathrm e}^{x} & \frac {{\mathrm e}^{3 x}}{9} & {\mathrm e}^{2 x} \left (\frac {3 \cos \left (x \right )}{25}+\frac {4 \sin \left (x \right )}{25}\right ) & {\mathrm e}^{2 x} \left (-\frac {3 \sin \left (x \right )}{25}+\frac {4 \cos \left (x \right )}{25}\right ) \\ {\mathrm e}^{x} & \frac {{\mathrm e}^{3 x}}{3} & {\mathrm e}^{2 x} \left (\frac {\sin \left (x \right )}{5}+\frac {2 \cos \left (x \right )}{5}\right ) & {\mathrm e}^{2 x} \left (-\frac {2 \sin \left (x \right )}{5}+\frac {\cos \left (x \right )}{5}\right ) \\ {\mathrm e}^{x} & {\mathrm e}^{3 x} & {\mathrm e}^{2 x} \cos \left (x \right ) & -{\mathrm e}^{2 x} \sin \left (x \right ) \end {array}\right ]\cdot \frac {1}{\left [\begin {array}{cccc} 1 & \frac {1}{27} & \frac {2}{125} & \frac {11}{125} \\ 1 & \frac {1}{9} & \frac {3}{25} & \frac {4}{25} \\ 1 & \frac {1}{3} & \frac {2}{5} & \frac {1}{5} \\ 1 & 1 & 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 {3 \left (2 \sin \left (x \right )-\cos \left (x \right )\right ) {\mathrm e}^{2 x}}{2}+\frac {15 \,{\mathrm e}^{x}}{4}-\frac {5 \,{\mathrm e}^{3 x}}{4} & \frac {\left (8 \cos \left (x \right )-22 \sin \left (x \right )\right ) {\mathrm e}^{2 x}}{4}-\frac {17 \,{\mathrm e}^{x}}{4}+\frac {9 \,{\mathrm e}^{3 x}}{4} & \frac {\left (-\cos \left (x \right )+6 \sin \left (x \right )\right ) {\mathrm e}^{2 x}}{2}+\frac {7 \,{\mathrm e}^{x}}{4}-\frac {5 \,{\mathrm e}^{3 x}}{4} & -\frac {{\mathrm e}^{2 x} \sin \left (x \right )}{2}-\frac {{\mathrm e}^{x}}{4}+\frac {{\mathrm e}^{3 x}}{4} \\ \frac {15 \,{\mathrm e}^{2 x} \sin \left (x \right )}{2}+\frac {15 \,{\mathrm e}^{x}}{4}-\frac {15 \,{\mathrm e}^{3 x}}{4} & \frac {\left (-6 \cos \left (x \right )-52 \sin \left (x \right )\right ) {\mathrm e}^{2 x}}{4}-\frac {17 \,{\mathrm e}^{x}}{4}+\frac {27 \,{\mathrm e}^{3 x}}{4} & \frac {\left (8 \cos \left (x \right )+26 \sin \left (x \right )\right ) {\mathrm e}^{2 x}}{4}+\frac {7 \,{\mathrm e}^{x}}{4}-\frac {15 \,{\mathrm e}^{3 x}}{4} & \frac {\left (-2 \cos \left (x \right )-4 \sin \left (x \right )\right ) {\mathrm e}^{2 x}}{4}-\frac {{\mathrm e}^{x}}{4}+\frac {3 \,{\mathrm e}^{3 x}}{4} \\ \frac {15 \left (\cos \left (x \right )+2 \sin \left (x \right )\right ) {\mathrm e}^{2 x}}{2}+\frac {15 \,{\mathrm e}^{x}}{4}-\frac {45 \,{\mathrm e}^{3 x}}{4} & \frac {\left (-64 \cos \left (x \right )-98 \sin \left (x \right )\right ) {\mathrm e}^{2 x}}{4}-\frac {17 \,{\mathrm e}^{x}}{4}+\frac {81 \,{\mathrm e}^{3 x}}{4} & \frac {\left (42 \cos \left (x \right )+44 \sin \left (x \right )\right ) {\mathrm e}^{2 x}}{4}+\frac {7 \,{\mathrm e}^{x}}{4}-\frac {45 \,{\mathrm e}^{3 x}}{4} & \frac {\left (-8 \cos \left (x \right )-6 \sin \left (x \right )\right ) {\mathrm e}^{2 x}}{4}-\frac {{\mathrm e}^{x}}{4}+\frac {9 \,{\mathrm e}^{3 x}}{4} \\ \frac {15 \left (4 \cos \left (x \right )+3 \sin \left (x \right )\right ) {\mathrm e}^{2 x}}{2}+\frac {15 \,{\mathrm e}^{x}}{4}-\frac {135 \,{\mathrm e}^{3 x}}{4} & \frac {\left (-113 \cos \left (x \right )-66 \sin \left (x \right )\right ) {\mathrm e}^{2 x}}{2}-\frac {17 \,{\mathrm e}^{x}}{4}+\frac {243 \,{\mathrm e}^{3 x}}{4} & \frac {\left (64 \cos \left (x \right )+23 \sin \left (x \right )\right ) {\mathrm e}^{2 x}}{2}+\frac {7 \,{\mathrm e}^{x}}{4}-\frac {135 \,{\mathrm e}^{3 x}}{4} & \frac {\left (-22 \cos \left (x \right )-4 \sin \left (x \right )\right ) {\mathrm e}^{2 x}}{4}-\frac {{\mathrm e}^{x}}{4}+\frac {27 \,{\mathrm e}^{3 x}}{4} \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 {{\mathrm e}^{x} \left (8 \,{\mathrm e}^{x} \cos \left (x \right )^{2} x -26 \sin \left (x \right ) {\mathrm e}^{x}-4 x \,{\mathrm e}^{x}+11 \,{\mathrm e}^{2 x}-11\right )}{4} \\ 4 \,{\mathrm e}^{x} \left (\frac {33 \,{\mathrm e}^{2 x}}{16}-\frac {11}{16}+\left (\left (\frac {1}{2}+x \right ) \cos \left (x \right )^{2}+\left (-\sin \left (x \right ) x -\frac {13}{8}\right ) \cos \left (x \right )-\frac {x}{2}-\frac {13 \sin \left (x \right )}{4}-\frac {1}{4}\right ) {\mathrm e}^{x}\right ) \\ -16 \,{\mathrm e}^{x} \left (-\frac {99 \,{\mathrm e}^{2 x}}{64}+\frac {11}{64}+\left (-\frac {\cos \left (x \right )^{2}}{2}+\left (\frac {13}{8}+\left (\frac {1}{2}+x \right ) \sin \left (x \right )\right ) \cos \left (x \right )+\frac {39 \sin \left (x \right )}{32}+\frac {1}{4}\right ) {\mathrm e}^{x}\right ) \\ -32 \,{\mathrm e}^{x} \left (-\frac {297 \,{\mathrm e}^{2 x}}{128}+\frac {11}{128}+\left (\cos \left (x \right )^{2} x +\left (\frac {143}{64}+\left (x +\frac {3}{2}\right ) \sin \left (x \right )\right ) \cos \left (x \right )-\frac {x}{2}+\frac {13 \sin \left (x \right )}{32}\right ) {\mathrm e}^{x}\right ) \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}+c_{2} {\moverset {\rightarrow }{y}}_{2}+c_{3} {\moverset {\rightarrow }{y}}_{3}\left (x \right )+c_{4} {\moverset {\rightarrow }{y}}_{4}\left (x \right )+\left [\begin {array}{c} \frac {{\mathrm e}^{x} \left (8 \,{\mathrm e}^{x} \cos \left (x \right )^{2} x -26 \sin \left (x \right ) {\mathrm e}^{x}-4 x \,{\mathrm e}^{x}+11 \,{\mathrm e}^{2 x}-11\right )}{4} \\ 4 \,{\mathrm e}^{x} \left (\frac {33 \,{\mathrm e}^{2 x}}{16}-\frac {11}{16}+\left (\left (\frac {1}{2}+x \right ) \cos \left (x \right )^{2}+\left (-\sin \left (x \right ) x -\frac {13}{8}\right ) \cos \left (x \right )-\frac {x}{2}-\frac {13 \sin \left (x \right )}{4}-\frac {1}{4}\right ) {\mathrm e}^{x}\right ) \\ -16 \,{\mathrm e}^{x} \left (-\frac {99 \,{\mathrm e}^{2 x}}{64}+\frac {11}{64}+\left (-\frac {\cos \left (x \right )^{2}}{2}+\left (\frac {13}{8}+\left (\frac {1}{2}+x \right ) \sin \left (x \right )\right ) \cos \left (x \right )+\frac {39 \sin \left (x \right )}{32}+\frac {1}{4}\right ) {\mathrm e}^{x}\right ) \\ -32 \,{\mathrm e}^{x} \left (-\frac {297 \,{\mathrm e}^{2 x}}{128}+\frac {11}{128}+\left (\cos \left (x \right )^{2} x +\left (\frac {143}{64}+\left (x +\frac {3}{2}\right ) \sin \left (x \right )\right ) \cos \left (x \right )-\frac {x}{2}+\frac {13 \sin \left (x \right )}{32}\right ) {\mathrm e}^{x}\right ) \end {array}\right ] \\ \bullet & {} & \textrm {First component of the vector is the solution to the ODE}\hspace {3pt} \\ {} & {} & y=\frac {\left (27000 \cos \left (x \right )^{2} x +\left (216 c_{3} +1188 c_{4} \right ) \cos \left (x \right )+\left (1188 c_{3} -216 c_{4} -87750\right ) \sin \left (x \right )-13500 x \right ) {\mathrm e}^{2 x}}{13500}+\frac {\left (500 c_{2} +37125\right ) {\mathrm e}^{3 x}}{13500}+{\mathrm e}^{x} \left (c_{1} -\frac {11}{4}\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`

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

\[ y(x)\to e^x \left (e^x x \cos (2 x)+c_4 e^{2 x}+c_2 e^x \cos (x)+c_1 e^x \sin (x)+c_3\right ) \]

19.39 problem section 9.3, problem 39

19.39.1 Maple step by step solution