problem Ex 15

Internal problem ID [11287]
Internal ﬁle name [OUTPUT/10273_Wednesday_December_21_2022_03_47_20_PM_39957394/index.tex]

Book: An elementary treatise on diﬀerential equations by Abraham Cohen. DC heath publishers. 1906
Section: Chapter VII, Linear diﬀerential equations with constant coeﬃcients. Article 52. Summary. Page 117
Problem number: Ex 15.
ODE order: 3.
ODE degree: 1.

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

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

Therefore the homogeneous solution is \[ y_h(x)=c_{1} {\mathrm e}^{x}+{\mathrm e}^{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x} c_{2} +{\mathrm e}^{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x} c_{3} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= {\mathrm e}^{x} \\ y_2 &= {\mathrm e}^{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x} \\ y_3 &= {\mathrm e}^{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\prime \prime \prime }-y = x \,{\mathrm e}^{x}+\cos \left (x \right )^{2} \] The particular solution is found using the method of undetermined coeﬃcients. Looking at the RHS of the ode, which is \[ x \,{\mathrm e}^{x}+\cos \left (x \right )^{2} \] Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is \[ [\{1\}, \{x \,{\mathrm e}^{x}, {\mathrm e}^{x}\}, \{\cos \left (2 x \right ), \sin \left (2 x \right )\}] \] While the set of the basis functions for the homogeneous solution found earlier is \[ \left \{{\mathrm e}^{x}, {\mathrm e}^{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x}, {\mathrm e}^{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x}\right \} \] Since \({\mathrm e}^{x}\) is duplicated in the UC_set, then this basis is multiplied by extra \(x\). The UC_set becomes \[ [\{1\}, \{x \,{\mathrm e}^{x}, {\mathrm e}^{x} x^{2}\}, \{\cos \left (2 x \right ), \sin \left (2 x \right )\}] \] Since there was duplication between the basis functions in the UC_set and the basis functions of the homogeneous solution, the trial solution is a linear combination of all the basis function in the above updated UC_set. \[ y_p = A_{1}+A_{2} x \,{\mathrm e}^{x}+A_{3} {\mathrm e}^{x} x^{2}+A_{4} \cos \left (2 x \right )+A_{5} \sin \left (2 x \right ) \] 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 \[ 3 A_{2} {\mathrm e}^{x}+6 A_{3} {\mathrm e}^{x} x +6 A_{3} {\mathrm e}^{x}+8 A_{4} \sin \left (2 x \right )-8 A_{5} \cos \left (2 x \right )-A_{1}-A_{4} \cos \left (2 x \right )-A_{5} \sin \left (2 x \right ) = x \,{\mathrm e}^{x}+\cos \left (x \right )^{2} \] Solving for the unknowns by comparing coeﬃcients results in \[ \left [A_{1} = -{\frac {1}{2}}, A_{2} = -{\frac {1}{3}}, A_{3} = {\frac {1}{6}}, A_{4} = -{\frac {1}{130}}, A_{5} = -{\frac {4}{65}}\right ] \] Substituting the above back in the above trial solution \(y_p\), gives the particular solution \[ y_p = -\frac {1}{2}-\frac {x \,{\mathrm e}^{x}}{3}+\frac {{\mathrm e}^{x} x^{2}}{6}-\frac {\cos \left (2 x \right )}{130}-\frac {4 \sin \left (2 x \right )}{65} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left (c_{1} {\mathrm e}^{x}+{\mathrm e}^{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x} c_{2} +{\mathrm e}^{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x} c_{3}\right ) + \left (-\frac {1}{2}-\frac {x \,{\mathrm e}^{x}}{3}+\frac {{\mathrm e}^{x} x^{2}}{6}-\frac {\cos \left (2 x \right )}{130}-\frac {4 \sin \left (2 x \right )}{65}\right ) \\ \end{align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} {\mathrm e}^{x}+{\mathrm e}^{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x} c_{2} +{\mathrm e}^{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x} c_{3} -\frac {1}{2}-\frac {x \,{\mathrm e}^{x}}{3}+\frac {{\mathrm e}^{x} x^{2}}{6}-\frac {\cos \left (2 x \right )}{130}-\frac {4 \sin \left (2 x \right )}{65} \\ \end{align*}

Veriﬁcation of solutions

\[ y = c_{1} {\mathrm e}^{x}+{\mathrm e}^{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x} c_{2} +{\mathrm e}^{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x} c_{3} -\frac {1}{2}-\frac {x \,{\mathrm e}^{x}}{3}+\frac {{\mathrm e}^{x} x^{2}}{6}-\frac {\cos \left (2 x \right )}{130}-\frac {4 \sin \left (2 x \right )}{65} \] Veriﬁed OK.

29.13.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime \prime \prime }-y=x \,{\mathrm e}^{x}+\cos \left (x \right )^{2} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 3 \\ {} & {} & y^{\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 {Isolate for}\hspace {3pt} y_{3}^{\prime }\left (x \right )\hspace {3pt}\textrm {using original ODE}\hspace {3pt} \\ {} & {} & y_{3}^{\prime }\left (x \right )=x \,{\mathrm e}^{x}+\cos \left (x \right )^{2}+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_{3}^{\prime }\left (x \right )=x \,{\mathrm e}^{x}+\cos \left (x \right )^{2}+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 ) \end {array}\right ] \\ \bullet & {} & \textrm {System to solve}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{y}}^{\prime }\left (x \right )=\left [\begin {array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end {array}\right ]\cdot {\moverset {\rightarrow }{y}}\left (x \right )+\left [\begin {array}{c} 0 \\ 0 \\ x \,{\mathrm e}^{x}+\cos \left (x \right )^{2} \end {array}\right ] \\ \bullet & {} & \textrm {Define the forcing function}\hspace {3pt} \\ {} & {} & {\moverset {\rightarrow }{f}}\left (x \right )=\left [\begin {array}{c} 0 \\ 0 \\ x \,{\mathrm e}^{x}+\cos \left (x \right )^{2} \end {array}\right ] \\ \bullet & {} & \textrm {Define the coefficient matrix}\hspace {3pt} \\ {} & {} & A =\left [\begin {array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 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, \left [\begin {array}{c} 1 \\ 1 \\ 1 \end {array}\right ]\right ], \left [-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}, \left [\begin {array}{c} \frac {1}{\left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right )^{2}} \\ \frac {1}{-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}} \\ 1 \end {array}\right ]\right ], \left [-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}, \left [\begin {array}{c} \frac {1}{\left (-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}\right )^{2}} \\ \frac {1}{-\frac {1}{2}+\frac {\mathrm {I} \sqrt {3}}{2}} \\ 1 \end {array}\right ]\right ]\right ] \\ \bullet & {} & \textrm {Consider eigenpair}\hspace {3pt} \\ {} & {} & \left [1, \left [\begin {array}{c} 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 \end {array}\right ] \\ \bullet & {} & \textrm {Consider complex eigenpair, complex conjugate eigenvalue can be ignored}\hspace {3pt} \\ {} & {} & \left [-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}, \left [\begin {array}{c} \frac {1}{\left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right )^{2}} \\ \frac {1}{-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution from eigenpair}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right ) x}\cdot \left [\begin {array}{c} \frac {1}{\left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right )^{2}} \\ \frac {1}{-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{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}^{-\frac {x}{2}}\cdot \left (\cos \left (\frac {\sqrt {3}\, x}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {3}\, x}{2}\right )\right )\cdot \left [\begin {array}{c} \frac {1}{\left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right )^{2}} \\ \frac {1}{-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Simplify expression}\hspace {3pt} \\ {} & {} & {\mathrm e}^{-\frac {x}{2}}\cdot \left [\begin {array}{c} \frac {\cos \left (\frac {\sqrt {3}\, x}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {3}\, x}{2}\right )}{\left (-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}\right )^{2}} \\ \frac {\cos \left (\frac {\sqrt {3}\, x}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {3}\, x}{2}\right )}{-\frac {1}{2}-\frac {\mathrm {I} \sqrt {3}}{2}} \\ \cos \left (\frac {\sqrt {3}\, x}{2}\right )-\mathrm {I} \sin \left (\frac {\sqrt {3}\, 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}}_{2}\left (x \right )={\mathrm e}^{-\frac {x}{2}}\cdot \left [\begin {array}{c} -\frac {\cos \left (\frac {\sqrt {3}\, x}{2}\right )}{2}-\frac {\sin \left (\frac {\sqrt {3}\, x}{2}\right ) \sqrt {3}}{2} \\ -\frac {\cos \left (\frac {\sqrt {3}\, x}{2}\right )}{2}+\frac {\sin \left (\frac {\sqrt {3}\, x}{2}\right ) \sqrt {3}}{2} \\ \cos \left (\frac {\sqrt {3}\, x}{2}\right ) \end {array}\right ], {\moverset {\rightarrow }{y}}_{3}\left (x \right )={\mathrm e}^{-\frac {x}{2}}\cdot \left [\begin {array}{c} -\frac {\cos \left (\frac {\sqrt {3}\, x}{2}\right ) \sqrt {3}}{2}+\frac {\sin \left (\frac {\sqrt {3}\, x}{2}\right )}{2} \\ \frac {\cos \left (\frac {\sqrt {3}\, x}{2}\right ) \sqrt {3}}{2}+\frac {\sin \left (\frac {\sqrt {3}\, x}{2}\right )}{2} \\ -\sin \left (\frac {\sqrt {3}\, 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}+c_{2} {\moverset {\rightarrow }{y}}_{2}\left (x \right )+c_{3} {\moverset {\rightarrow }{y}}_{3}\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}{ccc} {\mathrm e}^{x} & {\mathrm e}^{-\frac {x}{2}} \left (-\frac {\cos \left (\frac {\sqrt {3}\, x}{2}\right )}{2}-\frac {\sin \left (\frac {\sqrt {3}\, x}{2}\right ) \sqrt {3}}{2}\right ) & {\mathrm e}^{-\frac {x}{2}} \left (-\frac {\cos \left (\frac {\sqrt {3}\, x}{2}\right ) \sqrt {3}}{2}+\frac {\sin \left (\frac {\sqrt {3}\, x}{2}\right )}{2}\right ) \\ {\mathrm e}^{x} & {\mathrm e}^{-\frac {x}{2}} \left (-\frac {\cos \left (\frac {\sqrt {3}\, x}{2}\right )}{2}+\frac {\sin \left (\frac {\sqrt {3}\, x}{2}\right ) \sqrt {3}}{2}\right ) & {\mathrm e}^{-\frac {x}{2}} \left (\frac {\cos \left (\frac {\sqrt {3}\, x}{2}\right ) \sqrt {3}}{2}+\frac {\sin \left (\frac {\sqrt {3}\, x}{2}\right )}{2}\right ) \\ {\mathrm e}^{x} & {\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right ) & -{\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, 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}{ccc} {\mathrm e}^{x} & {\mathrm e}^{-\frac {x}{2}} \left (-\frac {\cos \left (\frac {\sqrt {3}\, x}{2}\right )}{2}-\frac {\sin \left (\frac {\sqrt {3}\, x}{2}\right ) \sqrt {3}}{2}\right ) & {\mathrm e}^{-\frac {x}{2}} \left (-\frac {\cos \left (\frac {\sqrt {3}\, x}{2}\right ) \sqrt {3}}{2}+\frac {\sin \left (\frac {\sqrt {3}\, x}{2}\right )}{2}\right ) \\ {\mathrm e}^{x} & {\mathrm e}^{-\frac {x}{2}} \left (-\frac {\cos \left (\frac {\sqrt {3}\, x}{2}\right )}{2}+\frac {\sin \left (\frac {\sqrt {3}\, x}{2}\right ) \sqrt {3}}{2}\right ) & {\mathrm e}^{-\frac {x}{2}} \left (\frac {\cos \left (\frac {\sqrt {3}\, x}{2}\right ) \sqrt {3}}{2}+\frac {\sin \left (\frac {\sqrt {3}\, x}{2}\right )}{2}\right ) \\ {\mathrm e}^{x} & {\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right ) & -{\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right ) \end {array}\right ]\cdot \frac {1}{\left [\begin {array}{ccc} 1 & -\frac {1}{2} & -\frac {\sqrt {3}}{2} \\ 1 & -\frac {1}{2} & \frac {\sqrt {3}}{2} \\ 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}{ccc} \frac {{\mathrm e}^{x}}{3}+\frac {2 \,{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )}{3} & \frac {{\mathrm e}^{x}}{3}-\frac {{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )}{3}+\frac {{\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right ) \sqrt {3}}{3} & \frac {{\mathrm e}^{x}}{3}-\frac {{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )}{3}-\frac {{\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right ) \sqrt {3}}{3} \\ \frac {{\mathrm e}^{x}}{3}-\frac {{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )}{3}-\frac {{\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right ) \sqrt {3}}{3} & \frac {{\mathrm e}^{x}}{3}+\frac {2 \,{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )}{3} & \frac {{\mathrm e}^{x}}{3}-\frac {{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )}{3}+\frac {{\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right ) \sqrt {3}}{3} \\ \frac {{\mathrm e}^{x}}{3}-\frac {{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )}{3}+\frac {{\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right ) \sqrt {3}}{3} & \frac {{\mathrm e}^{x}}{3}-\frac {{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )}{3}-\frac {{\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right ) \sqrt {3}}{3} & \frac {{\mathrm e}^{x}}{3}+\frac {2 \,{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )}{3} \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 {1}{2}+\frac {10 \,{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )}{117}+\frac {2 \,{\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right ) \sqrt {3}}{39}-\frac {\cos \left (2 x \right )}{130}-\frac {4 \sin \left (2 x \right )}{65}+\frac {\left (15 x^{2}-30 x +38\right ) {\mathrm e}^{x}}{90} \\ \frac {{\mathrm e}^{x} x^{2}}{6}-\frac {8 \,{\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right ) \sqrt {3}}{117}+\frac {4 \,{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )}{117}+\frac {4 \,{\mathrm e}^{x}}{45}-\frac {8 \cos \left (2 x \right )}{65}+\frac {\sin \left (2 x \right )}{65} \\ -\frac {14 \,{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )}{117}+\frac {2 \,{\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right ) \sqrt {3}}{117}+\frac {2 \cos \left (2 x \right )}{65}+\frac {16 \sin \left (2 x \right )}{65}+\frac {\left (15 x^{2}+30 x +8\right ) {\mathrm e}^{x}}{90} \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}\left (x \right )+c_{3} {\moverset {\rightarrow }{y}}_{3}\left (x \right )+\left [\begin {array}{c} -\frac {1}{2}+\frac {10 \,{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )}{117}+\frac {2 \,{\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right ) \sqrt {3}}{39}-\frac {\cos \left (2 x \right )}{130}-\frac {4 \sin \left (2 x \right )}{65}+\frac {\left (15 x^{2}-30 x +38\right ) {\mathrm e}^{x}}{90} \\ \frac {{\mathrm e}^{x} x^{2}}{6}-\frac {8 \,{\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right ) \sqrt {3}}{117}+\frac {4 \,{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )}{117}+\frac {4 \,{\mathrm e}^{x}}{45}-\frac {8 \cos \left (2 x \right )}{65}+\frac {\sin \left (2 x \right )}{65} \\ -\frac {14 \,{\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )}{117}+\frac {2 \,{\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right ) \sqrt {3}}{117}+\frac {2 \cos \left (2 x \right )}{65}+\frac {16 \sin \left (2 x \right )}{65}+\frac {\left (15 x^{2}+30 x +8\right ) {\mathrm e}^{x}}{90} \end {array}\right ] \\ \bullet & {} & \textrm {First component of the vector is the solution to the ODE}\hspace {3pt} \\ {} & {} & y=-\frac {1}{2}-\frac {{\mathrm e}^{-\frac {x}{2}} \left (\sqrt {3}\, c_{3} +c_{2} -\frac {20}{117}\right ) \cos \left (\frac {\sqrt {3}\, x}{2}\right )}{2}-\frac {{\mathrm e}^{-\frac {x}{2}} \left (\left (c_{2} -\frac {4}{39}\right ) \sqrt {3}-c_{3} \right ) \sin \left (\frac {\sqrt {3}\, x}{2}\right )}{2}-\frac {\cos \left (2 x \right )}{130}-\frac {4 \sin \left (2 x \right )}{65}+\frac {\left (15 x^{2}+90 c_{1} -30 x +38\right ) {\mathrm e}^{x}}{90} \end {array} \]

Maple trace

`Methods for third order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying high order exact linear fully integrable 
trying differential order: 3; linear nonhomogeneous with symmetry [0,1] 
trying high order linear exact nonhomogeneous 
trying differential order: 3; missing the dependent variable 
checking if the LODE has constant coefficients 
<- constant coefficients successful`

\[ y \left (x \right ) = -\frac {1}{2}+c_{2} {\mathrm e}^{-\frac {x}{2}} \cos \left (\frac {\sqrt {3}\, x}{2}\right )+c_{3} {\mathrm e}^{-\frac {x}{2}} \sin \left (\frac {\sqrt {3}\, x}{2}\right )-\frac {\cos \left (2 x \right )}{130}-\frac {4 \sin \left (2 x \right )}{65}+\frac {\left (3 x^{2}+18 c_{1} -6 x +4\right ) {\mathrm e}^{x}}{18} \]

\[ y(x)\to \frac {e^x x^2}{6}-\frac {e^x x}{3}+\frac {2 e^x}{9}-\frac {4}{65} \sin (2 x)-\frac {1}{130} \cos (2 x)+c_1 e^x+c_2 e^{-x/2} \cos \left (\frac {\sqrt {3} x}{2}\right )+c_3 e^{-x/2} \sin \left (\frac {\sqrt {3} x}{2}\right )-\frac {1}{2} \]

29.13 problem Ex 15

29.13.1 Maple step by step solution