15.69 problem 22.13 (g)

Internal problem ID [13763]
Internal file name [OUTPUT/12935_Sunday_February_18_2024_08_01_43_AM_6611512/index.tex]

Book: Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section: Chapter 22. Method of undetermined coefficients. Additional exercises page 412
Problem number: 22.13 (g).
ODE order: 3.
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

[[_3rd_order, _linear, _nonhomogeneous]]

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

Therefore the homogeneous solution is \[ y_h(x)=c_{1} {\mathrm e}^{x}+{\mathrm e}^{i x} c_{2} +{\mathrm e}^{-i x} c_{3} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= {\mathrm e}^{x} \\ y_2 &= {\mathrm e}^{i x} \\ y_3 &= {\mathrm e}^{-i x} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\prime \prime \prime }-y^{\prime \prime }+y^{\prime }-y = 5 x^{5} {\mathrm e}^{2 x} \] The particular solution is found using the method of undetermined coefficients. Looking at the RHS of the ode, which is \[ 5 x^{5} {\mathrm e}^{2 x} \] Shows that the corresponding undetermined set of the basis functions (UC_set) for the trial solution is \[ [\{x \,{\mathrm e}^{2 x}, x^{2} {\mathrm e}^{2 x}, x^{3} {\mathrm e}^{2 x}, x^{4} {\mathrm e}^{2 x}, x^{5} {\mathrm e}^{2 x}, {\mathrm e}^{2 x}\}] \] While the set of the basis functions for the homogeneous solution found earlier is \[ \{{\mathrm e}^{x}, {\mathrm e}^{i x}, {\mathrm e}^{-i x}\} \] 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. \[ y_p = A_{1} x \,{\mathrm e}^{2 x}+A_{2} x^{2} {\mathrm e}^{2 x}+A_{3} x^{3} {\mathrm e}^{2 x}+A_{4} x^{4} {\mathrm e}^{2 x}+A_{5} x^{5} {\mathrm e}^{2 x}+A_{6} {\mathrm e}^{2 x} \] The unknowns \(\{A_{1}, A_{2}, A_{3}, A_{4}, A_{5}, A_{6}\}\) 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 \[ 18 A_{2} x \,{\mathrm e}^{2 x}+27 A_{3} x^{2} {\mathrm e}^{2 x}+24 A_{4} x \,{\mathrm e}^{2 x}+60 A_{5} x^{2} {\mathrm e}^{2 x}+36 A_{4} x^{3} {\mathrm e}^{2 x}+45 A_{5} x^{4} {\mathrm e}^{2 x}+30 A_{3} x \,{\mathrm e}^{2 x}+60 A_{4} x^{2} {\mathrm e}^{2 x}+100 A_{5} x^{3} {\mathrm e}^{2 x}+5 A_{6} {\mathrm e}^{2 x}+5 A_{1} x \,{\mathrm e}^{2 x}+5 A_{2} x^{2} {\mathrm e}^{2 x}+5 A_{3} x^{3} {\mathrm e}^{2 x}+5 A_{4} x^{4} {\mathrm e}^{2 x}+5 A_{5} x^{5} {\mathrm e}^{2 x}+9 A_{1} {\mathrm e}^{2 x}+10 A_{2} {\mathrm e}^{2 x}+6 A_{3} {\mathrm e}^{2 x} = 5 x^{5} {\mathrm e}^{2 x} \] Solving for the unknowns by comparing coefficients results in \[ \left [A_{1} = {\frac {37464}{125}}, A_{2} = -{\frac {3648}{25}}, A_{3} = {\frac {224}{5}}, A_{4} = -9, A_{5} = 1, A_{6} = -{\frac {188376}{625}}\right ] \] Substituting the above back in the above trial solution \(y_p\), gives the particular solution \[ y_p = \frac {37464 x \,{\mathrm e}^{2 x}}{125}-\frac {3648 x^{2} {\mathrm e}^{2 x}}{25}+\frac {224 x^{3} {\mathrm e}^{2 x}}{5}-9 x^{4} {\mathrm e}^{2 x}+x^{5} {\mathrm e}^{2 x}-\frac {188376 \,{\mathrm e}^{2 x}}{625} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left (c_{1} {\mathrm e}^{x}+{\mathrm e}^{i x} c_{2} +{\mathrm e}^{-i x} c_{3}\right ) + \left (\frac {37464 x \,{\mathrm e}^{2 x}}{125}-\frac {3648 x^{2} {\mathrm e}^{2 x}}{25}+\frac {224 x^{3} {\mathrm e}^{2 x}}{5}-9 x^{4} {\mathrm e}^{2 x}+x^{5} {\mathrm e}^{2 x}-\frac {188376 \,{\mathrm e}^{2 x}}{625}\right ) \\ \end{align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} {\mathrm e}^{x}+{\mathrm e}^{i x} c_{2} +{\mathrm e}^{-i x} c_{3} +\frac {37464 x \,{\mathrm e}^{2 x}}{125}-\frac {3648 x^{2} {\mathrm e}^{2 x}}{25}+\frac {224 x^{3} {\mathrm e}^{2 x}}{5}-9 x^{4} {\mathrm e}^{2 x}+x^{5} {\mathrm e}^{2 x}-\frac {188376 \,{\mathrm e}^{2 x}}{625} \\ \end{align*}

Verification of solutions

\[ y = c_{1} {\mathrm e}^{x}+{\mathrm e}^{i x} c_{2} +{\mathrm e}^{-i x} c_{3} +\frac {37464 x \,{\mathrm e}^{2 x}}{125}-\frac {3648 x^{2} {\mathrm e}^{2 x}}{25}+\frac {224 x^{3} {\mathrm e}^{2 x}}{5}-9 x^{4} {\mathrm e}^{2 x}+x^{5} {\mathrm e}^{2 x}-\frac {188376 \,{\mathrm e}^{2 x}}{625} \] Verified OK.

15.69.1 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime \prime \prime }-y^{\prime \prime }+y^{\prime }-y=5 x^{5} {\mathrm e}^{2 x} \\ \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 )=5 x^{5} {\mathrm e}^{2 x}+y_{3}\left (x \right )-y_{2}\left (x \right )+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 )=5 x^{5} {\mathrm e}^{2 x}+y_{3}\left (x \right )-y_{2}\left (x \right )+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 & -1 & 1 \end {array}\right ]\cdot {\moverset {\rightarrow }{y}}\left (x \right )+\left [\begin {array}{c} 0 \\ 0 \\ 5 x^{5} {\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 \\ 5 x^{5} {\mathrm e}^{2 x} \end {array}\right ] \\ \bullet & {} & \textrm {Define the coefficient matrix}\hspace {3pt} \\ {} & {} & A =\left [\begin {array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & -1 & 1 \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 [\mathrm {-I}, \left [\begin {array}{c} -1 \\ \mathrm {I} \\ 1 \end {array}\right ]\right ], \left [\mathrm {I}, \left [\begin {array}{c} -1 \\ \mathrm {-I} \\ 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 [\mathrm {-I}, \left [\begin {array}{c} -1 \\ \mathrm {I} \\ 1 \end {array}\right ]\right ] \\ \bullet & {} & \textrm {Solution from eigenpair}\hspace {3pt} \\ {} & {} & {\mathrm e}^{\mathrm {-I} x}\cdot \left [\begin {array}{c} -1 \\ \mathrm {I} \\ 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 \\ {} & {} & \left (-\mathrm {I} \sin \left (x \right )+\cos \left (x \right )\right )\cdot \left [\begin {array}{c} -1 \\ \mathrm {I} \\ 1 \end {array}\right ] \\ \bullet & {} & \textrm {Simplify expression}\hspace {3pt} \\ {} & {} & \left [\begin {array}{c} -\cos \left (x \right )+\mathrm {I} \sin \left (x \right ) \\ \mathrm {I} \left (-\mathrm {I} \sin \left (x \right )+\cos \left (x \right )\right ) \\ -\mathrm {I} \sin \left (x \right )+\cos \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}}_{2}\left (x \right )=\left [\begin {array}{c} -\cos \left (x \right ) \\ \sin \left (x \right ) \\ \cos \left (x \right ) \end {array}\right ], {\moverset {\rightarrow }{y}}_{3}\left (x \right )=\left [\begin {array}{c} \sin \left (x \right ) \\ \cos \left (x \right ) \\ -\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}\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} & -\cos \left (x \right ) & \sin \left (x \right ) \\ {\mathrm e}^{x} & \sin \left (x \right ) & \cos \left (x \right ) \\ {\mathrm e}^{x} & \cos \left (x \right ) & -\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}{ccc} {\mathrm e}^{x} & -\cos \left (x \right ) & \sin \left (x \right ) \\ {\mathrm e}^{x} & \sin \left (x \right ) & \cos \left (x \right ) \\ {\mathrm e}^{x} & \cos \left (x \right ) & -\sin \left (x \right ) \end {array}\right ]\cdot \frac {1}{\left [\begin {array}{ccc} 1 & -1 & 0 \\ 1 & 0 & 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}{ccc} \frac {{\mathrm e}^{x}}{2}+\frac {\cos \left (x \right )}{2}-\frac {\sin \left (x \right )}{2} & \sin \left (x \right ) & \frac {{\mathrm e}^{x}}{2}-\frac {\cos \left (x \right )}{2}-\frac {\sin \left (x \right )}{2} \\ \frac {{\mathrm e}^{x}}{2}-\frac {\cos \left (x \right )}{2}-\frac {\sin \left (x \right )}{2} & \cos \left (x \right ) & \frac {{\mathrm e}^{x}}{2}+\frac {\sin \left (x \right )}{2}-\frac {\cos \left (x \right )}{2} \\ \frac {{\mathrm e}^{x}}{2}+\frac {\sin \left (x \right )}{2}-\frac {\cos \left (x \right )}{2} & -\sin \left (x \right ) & \frac {{\mathrm e}^{x}}{2}+\frac {\cos \left (x \right )}{2}+\frac {\sin \left (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 (625 x^{5}-5625 x^{4}+28000 x^{3}-91200 x^{2}+187320 x -188376\right ) {\mathrm e}^{2 x}}{625}+\frac {876 \cos \left (x \right )}{625}+300 \,{\mathrm e}^{x}+\frac {1932 \sin \left (x \right )}{625} \\ \frac {\left (1250 x^{5}-8125 x^{4}+33500 x^{3}-98400 x^{2}+192240 x -189432\right ) {\mathrm e}^{2 x}}{625}+\frac {1932 \cos \left (x \right )}{625}+300 \,{\mathrm e}^{x}-\frac {876 \sin \left (x \right )}{625} \\ \frac {4 \left (625 x^{5}-2500 x^{4}+8625 x^{3}-24075 x^{2}+46920 x -46656\right ) {\mathrm e}^{2 x}}{625}-\frac {876 \cos \left (x \right )}{625}+300 \,{\mathrm e}^{x}-\frac {1932 \sin \left (x \right )}{625} \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 {\left (625 x^{5}-5625 x^{4}+28000 x^{3}-91200 x^{2}+187320 x -188376\right ) {\mathrm e}^{2 x}}{625}+\frac {876 \cos \left (x \right )}{625}+300 \,{\mathrm e}^{x}+\frac {1932 \sin \left (x \right )}{625} \\ \frac {\left (1250 x^{5}-8125 x^{4}+33500 x^{3}-98400 x^{2}+192240 x -189432\right ) {\mathrm e}^{2 x}}{625}+\frac {1932 \cos \left (x \right )}{625}+300 \,{\mathrm e}^{x}-\frac {876 \sin \left (x \right )}{625} \\ \frac {4 \left (625 x^{5}-2500 x^{4}+8625 x^{3}-24075 x^{2}+46920 x -46656\right ) {\mathrm e}^{2 x}}{625}-\frac {876 \cos \left (x \right )}{625}+300 \,{\mathrm e}^{x}-\frac {1932 \sin \left (x \right )}{625} \end {array}\right ] \\ \bullet & {} & \textrm {First component of the vector is the solution to the ODE}\hspace {3pt} \\ {} & {} & y=\frac {\left (625 x^{5}-5625 x^{4}+28000 x^{3}-91200 x^{2}+187320 x -188376\right ) {\mathrm e}^{2 x}}{625}+\frac {\left (-625 c_{2} +876\right ) \cos \left (x \right )}{625}+\left (300+c_{1} \right ) {\mathrm e}^{x}+\frac {\sin \left (x \right ) \left (625 c_{3} +1932\right )}{625} \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`
 

✓ Solution by Maple

Time used: 0.0 (sec). Leaf size: 48

dsolve(diff(y(x),x$3)-diff(y(x),x$2)+diff(y(x),x)-y(x)=5*x^5*exp(2*x),y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {\left (625 x^{5}-5625 x^{4}+28000 x^{3}-91200 x^{2}+187320 x -188376\right ) {\mathrm e}^{2 x}}{625}+c_{1} \cos \left (x \right )+c_{2} {\mathrm e}^{x}+c_{3} \sin \left (x \right ) \]

✓ Solution by Mathematica

Time used: 0.009 (sec). Leaf size: 58

DSolve[y'''[x]-y''[x]+y'[x]-y[x]==5*x^5*Exp[2*x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to e^x \left (e^x \left (x^5-9 x^4+\frac {224 x^3}{5}-\frac {3648 x^2}{25}+\frac {37464 x}{125}-\frac {188376}{625}\right )+c_3\right )+c_1 \cos (x)+c_2 \sin (x) \]