11.50 problem 50

Internal problem ID [11823]
Internal file name [OUTPUT/11833_Thursday_April_11_2024_08_52_05_PM_42327415/index.tex]

Book: Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section: Chapter 4, Section 4.3. The method of undetermined coefficients. Exercises page 151
Problem number: 50.
ODE order: 6.
ODE degree: 1.

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

Maple gives the following as the ode type

[[_high_order, _missing_y]]

\[ \boxed {y^{\left (6\right )}+2 y^{\left (5\right )}+5 y^{\prime \prime \prime \prime }=x^{3}+x^{2} {\mathrm e}^{-x}+{\mathrm e}^{-x} \sin \left (2 x \right )} \] 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 \[ y^{\left (6\right )}+2 y^{\left (5\right )}+5 y^{\prime \prime \prime \prime } = 0 \] The characteristic equation is \[ \lambda ^{6}+2 \lambda ^{5}+5 \lambda ^{4} = 0 \] The roots of the above equation are \begin {align*} \lambda _1 &= 0\\ \lambda _2 &= 0\\ \lambda _3 &= 0\\ \lambda _4 &= 0\\ \lambda _5 &= -1+2 i\\ \lambda _6 &= -1-2 i \end {align*}

Therefore the homogeneous solution is \[ y_h(x)=x^{3} c_{4} +c_{3} x^{2}+c_{2} x +c_{1} +{\mathrm e}^{\left (-1+2 i\right ) x} c_{5} +{\mathrm e}^{\left (-1-2 i\right ) x} c_{6} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= 1 \\ y_2 &= x \\ y_3 &= x^{2} \\ y_4 &= x^{3} \\ y_5 &= {\mathrm e}^{\left (-1+2 i\right ) x} \\ y_6 &= {\mathrm e}^{\left (-1-2 i\right ) x} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\left (6\right )}+2 y^{\left (5\right )}+5 y^{\prime \prime \prime \prime } = x^{3}+x^{2} {\mathrm e}^{-x}+{\mathrm e}^{-x} \sin \left (2 x \right ) \] Let the particular solution be \[ y_p = U_1 y_1+U_2 y_2+U_3 y_3+U_4 y_4+U_5 y_5+U_6 y_6 \] Where \(y_i\) are the basis solutions found above for the homogeneous solution \(y_h\) and \(U_i(x)\) are functions to be determined as follows \[ U_i = (-1)^{n-i} \int { \frac {F(x) W_i(x) }{a W(x)} \, dx} \] Where \(W(x)\) is the Wronskian and \(W_i(x)\) is the Wronskian that results after deleting the last row and the \(i\)-th column of the determinant and \(n\) is the order of the ODE or equivalently, the number of basis solutions, and \(a\) is the coefficient of the leading derivative in the ODE, and \(F(x)\) is the RHS of the ODE. Therefore, the first step is to find the Wronskian \(W \left (x \right )\). This is given by \begin {equation*} W(x) = \begin {vmatrix} y_1&y_2&y_3&y_4&y_5&y_6\\ y_1'&y_2'&y_3'&y_4'&y_5'&y_6'\\ y_1''&y_2''&y_3''&y_4''&y_5''&y_6''\\ y_1'''&y_2'''&y_3'''&y_4'''&y_5'''&y_6'''\\ y_1''''&y_2''''&y_3''''&y_4''''&y_5''''&y_6''''\\ y_1'''''&y_2'''''&y_3'''''&y_4'''''&y_5'''''&y_6'''''\\ \end {vmatrix} \end {equation*} Substituting the fundamental set of solutions \(y_i\) found above in the Wronskian gives \begin {align*} W &= \left [\begin {array}{cccccc} 1 & x & x^{2} & x^{3} & {\mathrm e}^{\left (-1+2 i\right ) x} & {\mathrm e}^{\left (-1-2 i\right ) x} \\ 0 & 1 & 2 x & 3 x^{2} & \left (-1+2 i\right ) {\mathrm e}^{\left (-1+2 i\right ) x} & \left (-1-2 i\right ) {\mathrm e}^{\left (-1-2 i\right ) x} \\ 0 & 0 & 2 & 6 x & \left (-3-4 i\right ) {\mathrm e}^{\left (-1+2 i\right ) x} & \left (-3+4 i\right ) {\mathrm e}^{\left (-1-2 i\right ) x} \\ 0 & 0 & 0 & 6 & \left (11-2 i\right ) {\mathrm e}^{\left (-1+2 i\right ) x} & \left (11+2 i\right ) {\mathrm e}^{\left (-1-2 i\right ) x} \\ 0 & 0 & 0 & 0 & \left (-7+24 i\right ) {\mathrm e}^{\left (-1+2 i\right ) x} & \left (-7-24 i\right ) {\mathrm e}^{\left (-1-2 i\right ) x} \\ 0 & 0 & 0 & 0 & \left (-41-38 i\right ) {\mathrm e}^{\left (-1+2 i\right ) x} & \left (-41+38 i\right ) {\mathrm e}^{\left (-1-2 i\right ) x} \end {array}\right ] \\ |W| &= -30000 i {\mathrm e}^{\left (-1+2 i\right ) x} {\mathrm e}^{\left (-1-2 i\right ) x} \end {align*}

The determinant simplifies to \begin {align*} |W| &= -30000 i {\mathrm e}^{-2 x} \end {align*}

Now we determine \(W_i\) for each \(U_i\). \begin {align*} W_1(x) &= \det \,\left [\begin {array}{ccccc} x & x^{2} & x^{3} & {\mathrm e}^{\left (-1+2 i\right ) x} & {\mathrm e}^{\left (-1-2 i\right ) x} \\ 1 & 2 x & 3 x^{2} & \left (-1+2 i\right ) {\mathrm e}^{\left (-1+2 i\right ) x} & \left (-1-2 i\right ) {\mathrm e}^{\left (-1-2 i\right ) x} \\ 0 & 2 & 6 x & \left (-3-4 i\right ) {\mathrm e}^{\left (-1+2 i\right ) x} & \left (-3+4 i\right ) {\mathrm e}^{\left (-1-2 i\right ) x} \\ 0 & 0 & 6 & \left (11-2 i\right ) {\mathrm e}^{\left (-1+2 i\right ) x} & \left (11+2 i\right ) {\mathrm e}^{\left (-1-2 i\right ) x} \\ 0 & 0 & 0 & \left (-7+24 i\right ) {\mathrm e}^{\left (-1+2 i\right ) x} & \left (-7-24 i\right ) {\mathrm e}^{\left (-1-2 i\right ) x} \end {array}\right ] \\ &= -8 i {\mathrm e}^{-2 x} \left (125 x^{3}+150 x^{2}-30 x -72\right ) \end {align*}

\begin {align*} W_2(x) &= \det \,\left [\begin {array}{ccccc} 1 & x^{2} & x^{3} & {\mathrm e}^{\left (-1+2 i\right ) x} & {\mathrm e}^{\left (-1-2 i\right ) x} \\ 0 & 2 x & 3 x^{2} & \left (-1+2 i\right ) {\mathrm e}^{\left (-1+2 i\right ) x} & \left (-1-2 i\right ) {\mathrm e}^{\left (-1-2 i\right ) x} \\ 0 & 2 & 6 x & \left (-3-4 i\right ) {\mathrm e}^{\left (-1+2 i\right ) x} & \left (-3+4 i\right ) {\mathrm e}^{\left (-1-2 i\right ) x} \\ 0 & 0 & 6 & \left (11-2 i\right ) {\mathrm e}^{\left (-1+2 i\right ) x} & \left (11+2 i\right ) {\mathrm e}^{\left (-1-2 i\right ) x} \\ 0 & 0 & 0 & \left (-7+24 i\right ) {\mathrm e}^{\left (-1+2 i\right ) x} & \left (-7-24 i\right ) {\mathrm e}^{\left (-1-2 i\right ) x} \end {array}\right ] \\ &= -120 i {\mathrm e}^{-2 x} \left (25 x^{2}+20 x -2\right ) \end {align*}

\begin {align*} W_3(x) &= \det \,\left [\begin {array}{ccccc} 1 & x & x^{3} & {\mathrm e}^{\left (-1+2 i\right ) x} & {\mathrm e}^{\left (-1-2 i\right ) x} \\ 0 & 1 & 3 x^{2} & \left (-1+2 i\right ) {\mathrm e}^{\left (-1+2 i\right ) x} & \left (-1-2 i\right ) {\mathrm e}^{\left (-1-2 i\right ) x} \\ 0 & 0 & 6 x & \left (-3-4 i\right ) {\mathrm e}^{\left (-1+2 i\right ) x} & \left (-3+4 i\right ) {\mathrm e}^{\left (-1-2 i\right ) x} \\ 0 & 0 & 6 & \left (11-2 i\right ) {\mathrm e}^{\left (-1+2 i\right ) x} & \left (11+2 i\right ) {\mathrm e}^{\left (-1-2 i\right ) x} \\ 0 & 0 & 0 & \left (-7+24 i\right ) {\mathrm e}^{\left (-1+2 i\right ) x} & \left (-7-24 i\right ) {\mathrm e}^{\left (-1-2 i\right ) x} \end {array}\right ] \\ &= -600 i {\mathrm e}^{-2 x} \left (2+5 x \right ) \end {align*}

\begin {align*} W_4(x) &= \det \,\left [\begin {array}{ccccc} 1 & x & x^{2} & {\mathrm e}^{\left (-1+2 i\right ) x} & {\mathrm e}^{\left (-1-2 i\right ) x} \\ 0 & 1 & 2 x & \left (-1+2 i\right ) {\mathrm e}^{\left (-1+2 i\right ) x} & \left (-1-2 i\right ) {\mathrm e}^{\left (-1-2 i\right ) x} \\ 0 & 0 & 2 & \left (-3-4 i\right ) {\mathrm e}^{\left (-1+2 i\right ) x} & \left (-3+4 i\right ) {\mathrm e}^{\left (-1-2 i\right ) x} \\ 0 & 0 & 0 & \left (11-2 i\right ) {\mathrm e}^{\left (-1+2 i\right ) x} & \left (11+2 i\right ) {\mathrm e}^{\left (-1-2 i\right ) x} \\ 0 & 0 & 0 & \left (-7+24 i\right ) {\mathrm e}^{\left (-1+2 i\right ) x} & \left (-7-24 i\right ) {\mathrm e}^{\left (-1-2 i\right ) x} \end {array}\right ] \\ &= -1000 i {\mathrm e}^{-2 x} \end {align*}

\begin {align*} W_5(x) &= \det \,\left [\begin {array}{ccccc} 1 & x & x^{2} & x^{3} & {\mathrm e}^{\left (-1-2 i\right ) x} \\ 0 & 1 & 2 x & 3 x^{2} & \left (-1-2 i\right ) {\mathrm e}^{\left (-1-2 i\right ) x} \\ 0 & 0 & 2 & 6 x & \left (-3+4 i\right ) {\mathrm e}^{\left (-1-2 i\right ) x} \\ 0 & 0 & 0 & 6 & \left (11+2 i\right ) {\mathrm e}^{\left (-1-2 i\right ) x} \\ 0 & 0 & 0 & 0 & \left (-7-24 i\right ) {\mathrm e}^{\left (-1-2 i\right ) x} \end {array}\right ] \\ &= \left (-84-288 i\right ) {\mathrm e}^{\left (-1-2 i\right ) x} \end {align*}

\begin {align*} W_6(x) &= \det \,\left [\begin {array}{ccccc} 1 & x & x^{2} & x^{3} & {\mathrm e}^{\left (-1+2 i\right ) x} \\ 0 & 1 & 2 x & 3 x^{2} & \left (-1+2 i\right ) {\mathrm e}^{\left (-1+2 i\right ) x} \\ 0 & 0 & 2 & 6 x & \left (-3-4 i\right ) {\mathrm e}^{\left (-1+2 i\right ) x} \\ 0 & 0 & 0 & 6 & \left (11-2 i\right ) {\mathrm e}^{\left (-1+2 i\right ) x} \\ 0 & 0 & 0 & 0 & \left (-7+24 i\right ) {\mathrm e}^{\left (-1+2 i\right ) x} \end {array}\right ] \\ &= \left (-84+288 i\right ) {\mathrm e}^{\left (-1+2 i\right ) x} \end {align*}

Now we are ready to evaluate each \(U_i(x)\). \begin {align*} U_1 &= (-1)^{6-1} \int { \frac {F(x) W_1(x) }{a W(x)} \, dx}\\ &= (-1)^{5} \int { \frac { \left (x^{3}+x^{2} {\mathrm e}^{-x}+{\mathrm e}^{-x} \sin \left (2 x \right )\right ) \left (-8 i {\mathrm e}^{-2 x} \left (125 x^{3}+150 x^{2}-30 x -72\right )\right )}{\left (1\right ) \left (-30000 i {\mathrm e}^{-2 x}\right )} \, dx} \\ &= - \int { \frac {-8 i \left (x^{3}+x^{2} {\mathrm e}^{-x}+{\mathrm e}^{-x} \sin \left (2 x \right )\right ) {\mathrm e}^{-2 x} \left (125 x^{3}+150 x^{2}-30 x -72\right )}{-30000 i {\mathrm e}^{-2 x}} \, dx}\\ &= - \int {\left (\frac {\left (x^{3}+x^{2} {\mathrm e}^{-x}+{\mathrm e}^{-x} \sin \left (2 x \right )\right ) \left (125 x^{3}+150 x^{2}-30 x -72\right )}{3750}\right ) \, dx}\\ &= \frac {3 x^{4}}{625}+\frac {3046 \,{\mathrm e}^{-x}}{625}+\frac {x^{5}}{625}+\frac {31 \,{\mathrm e}^{-x} x^{4}}{150}-\frac {x^{6}}{150}+\frac {3046 x \,{\mathrm e}^{-x}}{625}+\frac {1523 x^{2} {\mathrm e}^{-x}}{625}+\frac {307 \,{\mathrm e}^{-x} x^{3}}{375}+\frac {12 \,{\mathrm e}^{-x} \left (-\sin \left (2 x \right )-2 \cos \left (2 x \right )\right )}{3125}+\frac {{\mathrm e}^{-x} x^{5}}{30}-\frac {\left (-\frac {2}{5} x^{2}-\frac {8}{25} x +\frac {4}{125}\right ) {\mathrm e}^{-x} \cos \left (2 x \right )}{25}-\frac {\left (-\frac {1}{5} x^{2}+\frac {6}{25} x +\frac {22}{125}\right ) {\mathrm e}^{-x} \sin \left (2 x \right )}{25}+\frac {\left (-\frac {2 x}{5}-\frac {4}{25}\right ) {\mathrm e}^{-x} \cos \left (2 x \right )}{125}+\frac {\left (-\frac {x}{5}+\frac {3}{25}\right ) {\mathrm e}^{-x} \sin \left (2 x \right )}{125}-\frac {\left (-\frac {2}{5} x^{3}-\frac {12}{25} x^{2}+\frac {12}{125} x +\frac {144}{625}\right ) {\mathrm e}^{-x} \cos \left (2 x \right )}{30}-\frac {\left (-\frac {1}{5} x^{3}+\frac {9}{25} x^{2}+\frac {66}{125} x +\frac {42}{625}\right ) {\mathrm e}^{-x} \sin \left (2 x \right )}{30}-\frac {x^{7}}{210} \end {align*}

\begin {align*} U_2 &= (-1)^{6-2} \int { \frac {F(x) W_2(x) }{a W(x)} \, dx}\\ &= (-1)^{4} \int { \frac { \left (x^{3}+x^{2} {\mathrm e}^{-x}+{\mathrm e}^{-x} \sin \left (2 x \right )\right ) \left (-120 i {\mathrm e}^{-2 x} \left (25 x^{2}+20 x -2\right )\right )}{\left (1\right ) \left (-30000 i {\mathrm e}^{-2 x}\right )} \, dx} \\ &= \int { \frac {-120 i \left (x^{3}+x^{2} {\mathrm e}^{-x}+{\mathrm e}^{-x} \sin \left (2 x \right )\right ) {\mathrm e}^{-2 x} \left (25 x^{2}+20 x -2\right )}{-30000 i {\mathrm e}^{-2 x}} \, dx}\\ &= \int {\left (\frac {\left (x^{3}+x^{2} {\mathrm e}^{-x}+{\mathrm e}^{-x} \sin \left (2 x \right )\right ) \left (25 x^{2}+20 x -2\right )}{250}\right ) \, dx}\\ &= \frac {x^{6}}{60}+\frac {2 x^{5}}{125}-\frac {x^{4}}{500}-\frac {{\mathrm e}^{-x} x^{4}}{10}-\frac {12 \,{\mathrm e}^{-x} x^{3}}{25}-\frac {179 x^{2} {\mathrm e}^{-x}}{125}-\frac {358 x \,{\mathrm e}^{-x}}{125}-\frac {358 \,{\mathrm e}^{-x}}{125}+\frac {\left (-\frac {2}{5} x^{2}-\frac {8}{25} x +\frac {4}{125}\right ) {\mathrm e}^{-x} \cos \left (2 x \right )}{10}+\frac {\left (-\frac {1}{5} x^{2}+\frac {6}{25} x +\frac {22}{125}\right ) {\mathrm e}^{-x} \sin \left (2 x \right )}{10}+\frac {2 \left (-\frac {2 x}{5}-\frac {4}{25}\right ) {\mathrm e}^{-x} \cos \left (2 x \right )}{25}+\frac {2 \left (-\frac {x}{5}+\frac {3}{25}\right ) {\mathrm e}^{-x} \sin \left (2 x \right )}{25}-\frac {{\mathrm e}^{-x} \left (-\sin \left (2 x \right )-2 \cos \left (2 x \right )\right )}{625} \end {align*}

\begin {align*} U_3 &= (-1)^{6-3} \int { \frac {F(x) W_3(x) }{a W(x)} \, dx}\\ &= (-1)^{3} \int { \frac { \left (x^{3}+x^{2} {\mathrm e}^{-x}+{\mathrm e}^{-x} \sin \left (2 x \right )\right ) \left (-600 i {\mathrm e}^{-2 x} \left (2+5 x \right )\right )}{\left (1\right ) \left (-30000 i {\mathrm e}^{-2 x}\right )} \, dx} \\ &= - \int { \frac {-600 i \left (x^{3}+x^{2} {\mathrm e}^{-x}+{\mathrm e}^{-x} \sin \left (2 x \right )\right ) {\mathrm e}^{-2 x} \left (2+5 x \right )}{-30000 i {\mathrm e}^{-2 x}} \, dx}\\ &= - \int {\left (\frac {\left (\frac {2}{5}+x \right ) \left (\left (x^{2}+\sin \left (2 x \right )\right ) {\mathrm e}^{-x}+x^{3}\right )}{10}\right ) \, dx} \\ &= -\frac {x^{4}}{100}+\frac {17 x^{2} {\mathrm e}^{-x}}{50}+\frac {17 x \,{\mathrm e}^{-x}}{25}+\frac {17 \,{\mathrm e}^{-x}}{25}-\frac {{\mathrm e}^{-x} \left (-\sin \left (2 x \right )-2 \cos \left (2 x \right )\right )}{125}-\frac {x^{5}}{50}+\frac {{\mathrm e}^{-x} x^{3}}{10}-\frac {\left (-\frac {2 x}{5}-\frac {4}{25}\right ) {\mathrm e}^{-x} \cos \left (2 x \right )}{10}-\frac {\left (-\frac {x}{5}+\frac {3}{25}\right ) {\mathrm e}^{-x} \sin \left (2 x \right )}{10} \\ &= -\frac {x^{4}}{100}+\frac {17 x^{2} {\mathrm e}^{-x}}{50}+\frac {17 x \,{\mathrm e}^{-x}}{25}+\frac {17 \,{\mathrm e}^{-x}}{25}-\frac {{\mathrm e}^{-x} \left (-\sin \left (2 x \right )-2 \cos \left (2 x \right )\right )}{125}-\frac {x^{5}}{50}+\frac {{\mathrm e}^{-x} x^{3}}{10}-\frac {\left (-\frac {2 x}{5}-\frac {4}{25}\right ) {\mathrm e}^{-x} \cos \left (2 x \right )}{10}-\frac {\left (-\frac {x}{5}+\frac {3}{25}\right ) {\mathrm e}^{-x} \sin \left (2 x \right )}{10} \end {align*}

\begin {align*} U_4 &= (-1)^{6-4} \int { \frac {F(x) W_4(x) }{a W(x)} \, dx}\\ &= (-1)^{2} \int { \frac { \left (x^{3}+x^{2} {\mathrm e}^{-x}+{\mathrm e}^{-x} \sin \left (2 x \right )\right ) \left (-1000 i {\mathrm e}^{-2 x}\right )}{\left (1\right ) \left (-30000 i {\mathrm e}^{-2 x}\right )} \, dx} \\ &= \int { \frac {-1000 i \left (x^{3}+x^{2} {\mathrm e}^{-x}+{\mathrm e}^{-x} \sin \left (2 x \right )\right ) {\mathrm e}^{-2 x}}{-30000 i {\mathrm e}^{-2 x}} \, dx}\\ &= \int {\left (\frac {\left (x^{2}+\sin \left (2 x \right )\right ) {\mathrm e}^{-x}}{30}+\frac {x^{3}}{30}\right ) \, dx} \\ &= \frac {x^{4}}{120}-\frac {x^{2} {\mathrm e}^{-x}}{30}-\frac {x \,{\mathrm e}^{-x}}{15}-\frac {{\mathrm e}^{-x}}{15}+\frac {{\mathrm e}^{-x} \left (-\sin \left (2 x \right )-2 \cos \left (2 x \right )\right )}{150} \\ &= \frac {x^{4}}{120}-\frac {x^{2} {\mathrm e}^{-x}}{30}-\frac {x \,{\mathrm e}^{-x}}{15}-\frac {{\mathrm e}^{-x}}{15}+\frac {{\mathrm e}^{-x} \left (-\sin \left (2 x \right )-2 \cos \left (2 x \right )\right )}{150} \end {align*}

\begin {align*} U_5 &= (-1)^{6-5} \int { \frac {F(x) W_5(x) }{a W(x)} \, dx}\\ &= (-1)^{1} \int { \frac { \left (x^{3}+x^{2} {\mathrm e}^{-x}+{\mathrm e}^{-x} \sin \left (2 x \right )\right ) \left (\left (-84-288 i\right ) {\mathrm e}^{\left (-1-2 i\right ) x}\right )}{\left (1\right ) \left (-30000 i {\mathrm e}^{-2 x}\right )} \, dx} \\ &= - \int { \frac {\left (-84-288 i\right ) \left (x^{3}+x^{2} {\mathrm e}^{-x}+{\mathrm e}^{-x} \sin \left (2 x \right )\right ) {\mathrm e}^{\left (-1-2 i\right ) x}}{-30000 i {\mathrm e}^{-2 x}} \, dx}\\ &= - \int {\left (\left (\frac {6}{625}-\frac {7 i}{2500}\right ) {\mathrm e}^{\left (1-2 i\right ) x} \left (\left (x^{2}+\sin \left (2 x \right )\right ) {\mathrm e}^{-x}+x^{3}\right )\right ) \, dx} \\ &= \left (-\frac {6}{625}+\frac {7 i}{2500}\right ) \left (\left (-\frac {7}{625}-\frac {24 i}{625}\right ) \left (\left (-11+2 i\right ) {\mathrm e}^{\left (1-2 i\right ) x} x^{3}+\left (9+12 i\right ) x^{2} {\mathrm e}^{\left (1-2 i\right ) x}+\left (6-12 i\right ) {\mathrm e}^{\left (1-2 i\right ) x} x -6 \,{\mathrm e}^{\left (1-2 i\right ) x}\right )+i \left (\frac {{\mathrm e}^{-2 i x} x^{2}}{2}-\frac {i {\mathrm e}^{-2 i x} x}{2}-\frac {{\mathrm e}^{-2 i x}}{4}\right )+\frac {\frac {i {\mathrm e}^{-x} {\mathrm e}^{\left (1-2 i\right ) x} \tan \left (x \right )}{2}+{\mathrm e}^{-x} x \,{\mathrm e}^{\left (1-2 i\right ) x} \tan \left (x \right )-\frac {i x \,{\mathrm e}^{-x} {\mathrm e}^{\left (1-2 i\right ) x}}{2}+\frac {i x \,{\mathrm e}^{-x} {\mathrm e}^{\left (1-2 i\right ) x} \tan \left (x \right )^{2}}{2}}{1+\tan \left (x \right )^{2}}\right ) \\ &= \left (-\frac {6}{625}+\frac {7 i}{2500}\right ) \left (\left (-\frac {7}{625}-\frac {24 i}{625}\right ) \left (\left (-11+2 i\right ) {\mathrm e}^{\left (1-2 i\right ) x} x^{3}+\left (9+12 i\right ) x^{2} {\mathrm e}^{\left (1-2 i\right ) x}+\left (6-12 i\right ) {\mathrm e}^{\left (1-2 i\right ) x} x -6 \,{\mathrm e}^{\left (1-2 i\right ) x}\right )+i \left (\frac {{\mathrm e}^{-2 i x} x^{2}}{2}-\frac {i {\mathrm e}^{-2 i x} x}{2}-\frac {{\mathrm e}^{-2 i x}}{4}\right )+\frac {\frac {i {\mathrm e}^{-x} {\mathrm e}^{\left (1-2 i\right ) x} \tan \left (x \right )}{2}+{\mathrm e}^{-x} x \,{\mathrm e}^{\left (1-2 i\right ) x} \tan \left (x \right )-\frac {i x \,{\mathrm e}^{-x} {\mathrm e}^{\left (1-2 i\right ) x}}{2}+\frac {i x \,{\mathrm e}^{-x} {\mathrm e}^{\left (1-2 i\right ) x} \tan \left (x \right )^{2}}{2}}{1+\tan \left (x \right )^{2}}\right ) \end {align*}

\begin {align*} U_6 &= (-1)^{6-6} \int { \frac {F(x) W_6(x) }{a W(x)} \, dx}\\ &= (-1)^{0} \int { \frac { \left (x^{3}+x^{2} {\mathrm e}^{-x}+{\mathrm e}^{-x} \sin \left (2 x \right )\right ) \left (\left (-84+288 i\right ) {\mathrm e}^{\left (-1+2 i\right ) x}\right )}{\left (1\right ) \left (-30000 i {\mathrm e}^{-2 x}\right )} \, dx} \\ &= \int { \frac {\left (-84+288 i\right ) \left (x^{3}+x^{2} {\mathrm e}^{-x}+{\mathrm e}^{-x} \sin \left (2 x \right )\right ) {\mathrm e}^{\left (-1+2 i\right ) x}}{-30000 i {\mathrm e}^{-2 x}} \, dx}\\ &= \int {\left (\left (-\frac {6}{625}-\frac {7 i}{2500}\right ) {\mathrm e}^{\left (1+2 i\right ) x} \left (\left (x^{2}+\sin \left (2 x \right )\right ) {\mathrm e}^{-x}+x^{3}\right )\right ) \, dx} \\ &= \left (-\frac {6}{625}-\frac {7 i}{2500}\right ) \left (\left (-\frac {7}{625}+\frac {24 i}{625}\right ) \left (\left (-11-2 i\right ) {\mathrm e}^{\left (1+2 i\right ) x} x^{3}+\left (9-12 i\right ) x^{2} {\mathrm e}^{\left (1+2 i\right ) x}+\left (6+12 i\right ) {\mathrm e}^{\left (1+2 i\right ) x} x -6 \,{\mathrm e}^{\left (1+2 i\right ) x}\right )+i \left (-\frac {{\mathrm e}^{2 i x} x^{2}}{2}-\frac {i x \,{\mathrm e}^{2 i x}}{2}+\frac {{\mathrm e}^{2 i x}}{4}\right )+\frac {-\frac {i {\mathrm e}^{-x} {\mathrm e}^{\left (1+2 i\right ) x} \tan \left (x \right )}{2}+{\mathrm e}^{-x} x \,{\mathrm e}^{\left (1+2 i\right ) x} \tan \left (x \right )+\frac {i x \,{\mathrm e}^{-x} {\mathrm e}^{\left (1+2 i\right ) x}}{2}-\frac {i x \,{\mathrm e}^{-x} {\mathrm e}^{\left (1+2 i\right ) x} \tan \left (x \right )^{2}}{2}}{1+\tan \left (x \right )^{2}}\right ) \\ &= \left (-\frac {6}{625}-\frac {7 i}{2500}\right ) \left (\left (-\frac {7}{625}+\frac {24 i}{625}\right ) \left (\left (-11-2 i\right ) {\mathrm e}^{\left (1+2 i\right ) x} x^{3}+\left (9-12 i\right ) x^{2} {\mathrm e}^{\left (1+2 i\right ) x}+\left (6+12 i\right ) {\mathrm e}^{\left (1+2 i\right ) x} x -6 \,{\mathrm e}^{\left (1+2 i\right ) x}\right )+i \left (-\frac {{\mathrm e}^{2 i x} x^{2}}{2}-\frac {i x \,{\mathrm e}^{2 i x}}{2}+\frac {{\mathrm e}^{2 i x}}{4}\right )+\frac {-\frac {i {\mathrm e}^{-x} {\mathrm e}^{\left (1+2 i\right ) x} \tan \left (x \right )}{2}+{\mathrm e}^{-x} x \,{\mathrm e}^{\left (1+2 i\right ) x} \tan \left (x \right )+\frac {i x \,{\mathrm e}^{-x} {\mathrm e}^{\left (1+2 i\right ) x}}{2}-\frac {i x \,{\mathrm e}^{-x} {\mathrm e}^{\left (1+2 i\right ) x} \tan \left (x \right )^{2}}{2}}{1+\tan \left (x \right )^{2}}\right ) \end {align*}

Now that all the \(U_i\) functions have been determined, the particular solution is found from \[ y_p = U_1 y_1+U_2 y_2+U_3 y_3+U_4 y_4+U_5 y_5+U_6 y_6 \] Hence \begin {equation*} \begin {split} y_p &= \left (\frac {3 x^{4}}{625}+\frac {3046 \,{\mathrm e}^{-x}}{625}+\frac {x^{5}}{625}+\frac {31 \,{\mathrm e}^{-x} x^{4}}{150}-\frac {x^{6}}{150}+\frac {3046 x \,{\mathrm e}^{-x}}{625}+\frac {1523 x^{2} {\mathrm e}^{-x}}{625}+\frac {307 \,{\mathrm e}^{-x} x^{3}}{375}+\frac {12 \,{\mathrm e}^{-x} \left (-\sin \left (2 x \right )-2 \cos \left (2 x \right )\right )}{3125}+\frac {{\mathrm e}^{-x} x^{5}}{30}-\frac {\left (-\frac {2}{5} x^{2}-\frac {8}{25} x +\frac {4}{125}\right ) {\mathrm e}^{-x} \cos \left (2 x \right )}{25}-\frac {\left (-\frac {1}{5} x^{2}+\frac {6}{25} x +\frac {22}{125}\right ) {\mathrm e}^{-x} \sin \left (2 x \right )}{25}+\frac {\left (-\frac {2 x}{5}-\frac {4}{25}\right ) {\mathrm e}^{-x} \cos \left (2 x \right )}{125}+\frac {\left (-\frac {x}{5}+\frac {3}{25}\right ) {\mathrm e}^{-x} \sin \left (2 x \right )}{125}-\frac {\left (-\frac {2}{5} x^{3}-\frac {12}{25} x^{2}+\frac {12}{125} x +\frac {144}{625}\right ) {\mathrm e}^{-x} \cos \left (2 x \right )}{30}-\frac {\left (-\frac {1}{5} x^{3}+\frac {9}{25} x^{2}+\frac {66}{125} x +\frac {42}{625}\right ) {\mathrm e}^{-x} \sin \left (2 x \right )}{30}-\frac {x^{7}}{210}\right ) \\ &+\left (\frac {x^{6}}{60}+\frac {2 x^{5}}{125}-\frac {x^{4}}{500}-\frac {{\mathrm e}^{-x} x^{4}}{10}-\frac {12 \,{\mathrm e}^{-x} x^{3}}{25}-\frac {179 x^{2} {\mathrm e}^{-x}}{125}-\frac {358 x \,{\mathrm e}^{-x}}{125}-\frac {358 \,{\mathrm e}^{-x}}{125}+\frac {\left (-\frac {2}{5} x^{2}-\frac {8}{25} x +\frac {4}{125}\right ) {\mathrm e}^{-x} \cos \left (2 x \right )}{10}+\frac {\left (-\frac {1}{5} x^{2}+\frac {6}{25} x +\frac {22}{125}\right ) {\mathrm e}^{-x} \sin \left (2 x \right )}{10}+\frac {2 \left (-\frac {2 x}{5}-\frac {4}{25}\right ) {\mathrm e}^{-x} \cos \left (2 x \right )}{25}+\frac {2 \left (-\frac {x}{5}+\frac {3}{25}\right ) {\mathrm e}^{-x} \sin \left (2 x \right )}{25}-\frac {{\mathrm e}^{-x} \left (-\sin \left (2 x \right )-2 \cos \left (2 x \right )\right )}{625}\right ) \left (x\right ) \\ &+\left (-\frac {x^{4}}{100}+\frac {17 x^{2} {\mathrm e}^{-x}}{50}+\frac {17 x \,{\mathrm e}^{-x}}{25}+\frac {17 \,{\mathrm e}^{-x}}{25}-\frac {{\mathrm e}^{-x} \left (-\sin \left (2 x \right )-2 \cos \left (2 x \right )\right )}{125}-\frac {x^{5}}{50}+\frac {{\mathrm e}^{-x} x^{3}}{10}-\frac {\left (-\frac {2 x}{5}-\frac {4}{25}\right ) {\mathrm e}^{-x} \cos \left (2 x \right )}{10}-\frac {\left (-\frac {x}{5}+\frac {3}{25}\right ) {\mathrm e}^{-x} \sin \left (2 x \right )}{10}\right ) \left (x^{2}\right ) \\ &+\left (\frac {x^{4}}{120}-\frac {x^{2} {\mathrm e}^{-x}}{30}-\frac {x \,{\mathrm e}^{-x}}{15}-\frac {{\mathrm e}^{-x}}{15}+\frac {{\mathrm e}^{-x} \left (-\sin \left (2 x \right )-2 \cos \left (2 x \right )\right )}{150}\right ) \left (x^{3}\right ) \\ &+\left (\left (-\frac {6}{625}+\frac {7 i}{2500}\right ) \left (\left (-\frac {7}{625}-\frac {24 i}{625}\right ) \left (\left (-11+2 i\right ) {\mathrm e}^{\left (1-2 i\right ) x} x^{3}+\left (9+12 i\right ) x^{2} {\mathrm e}^{\left (1-2 i\right ) x}+\left (6-12 i\right ) {\mathrm e}^{\left (1-2 i\right ) x} x -6 \,{\mathrm e}^{\left (1-2 i\right ) x}\right )+i \left (\frac {{\mathrm e}^{-2 i x} x^{2}}{2}-\frac {i {\mathrm e}^{-2 i x} x}{2}-\frac {{\mathrm e}^{-2 i x}}{4}\right )+\frac {\frac {i {\mathrm e}^{-x} {\mathrm e}^{\left (1-2 i\right ) x} \tan \left (x \right )}{2}+{\mathrm e}^{-x} x \,{\mathrm e}^{\left (1-2 i\right ) x} \tan \left (x \right )-\frac {i x \,{\mathrm e}^{-x} {\mathrm e}^{\left (1-2 i\right ) x}}{2}+\frac {i x \,{\mathrm e}^{-x} {\mathrm e}^{\left (1-2 i\right ) x} \tan \left (x \right )^{2}}{2}}{1+\tan \left (x \right )^{2}}\right )\right ) \left ({\mathrm e}^{\left (-1+2 i\right ) x}\right ) \\ &+\left (\left (-\frac {6}{625}-\frac {7 i}{2500}\right ) \left (\left (-\frac {7}{625}+\frac {24 i}{625}\right ) \left (\left (-11-2 i\right ) {\mathrm e}^{\left (1+2 i\right ) x} x^{3}+\left (9-12 i\right ) x^{2} {\mathrm e}^{\left (1+2 i\right ) x}+\left (6+12 i\right ) {\mathrm e}^{\left (1+2 i\right ) x} x -6 \,{\mathrm e}^{\left (1+2 i\right ) x}\right )+i \left (-\frac {{\mathrm e}^{2 i x} x^{2}}{2}-\frac {i x \,{\mathrm e}^{2 i x}}{2}+\frac {{\mathrm e}^{2 i x}}{4}\right )+\frac {-\frac {i {\mathrm e}^{-x} {\mathrm e}^{\left (1+2 i\right ) x} \tan \left (x \right )}{2}+{\mathrm e}^{-x} x \,{\mathrm e}^{\left (1+2 i\right ) x} \tan \left (x \right )+\frac {i x \,{\mathrm e}^{-x} {\mathrm e}^{\left (1+2 i\right ) x}}{2}-\frac {i x \,{\mathrm e}^{-x} {\mathrm e}^{\left (1+2 i\right ) x} \tan \left (x \right )^{2}}{2}}{1+\tan \left (x \right )^{2}}\right )\right ) \left ({\mathrm e}^{\left (-1-2 i\right ) x}\right ) \end {split} \end {equation*} Therefore the particular solution is \[ y_p = -\frac {1008}{390625}+\frac {\left (-448-339 i+\left (70-240 i\right ) x \right ) {\mathrm e}^{\left (-1-2 i\right ) x}}{50000}+\frac {\left (-448+339 i+\left (70+240 i\right ) x \right ) {\mathrm e}^{\left (-1+2 i\right ) x}}{50000}+\frac {\left (2 x^{2}+16 x +39\right ) {\mathrm e}^{-x}}{8}+\frac {x^{7}}{4200}-\frac {x^{6}}{1500}-\frac {x^{5}}{2500}+\frac {3 x^{4}}{625}-\frac {19 x^{3}}{3125}-\frac {66 x^{2}}{15625}+\frac {834 x}{78125} \] Which simplifies to \[ y_p = \frac {\left (\left (183750 x -1176000\right ) \cos \left (2 x \right )+\left (-630000 x -889875\right ) \sin \left (2 x \right )[phone] x^{2}[phone] x [phone]\right ) {\mathrm e}^{-x}}{65625000}+\frac {x^{7}}{4200}-\frac {x^{6}}{1500}-\frac {x^{5}}{2500}+\frac {3 x^{4}}{625}-\frac {19 x^{3}}{3125}-\frac {66 x^{2}}{15625}+\frac {834 x}{78125}-\frac {1008}{390625} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left (x^{3} c_{4} +c_{3} x^{2}+c_{2} x +c_{1} +{\mathrm e}^{\left (-1+2 i\right ) x} c_{5} +{\mathrm e}^{\left (-1-2 i\right ) x} c_{6}\right ) + \left (\frac {\left (\left (183750 x -1176000\right ) \cos \left (2 x \right )+\left (-630000 x -889875\right ) \sin \left (2 x \right )[phone] x^{2}[phone] x [phone]\right ) {\mathrm e}^{-x}}{65625000}+\frac {x^{7}}{4200}-\frac {x^{6}}{1500}-\frac {x^{5}}{2500}+\frac {3 x^{4}}{625}-\frac {19 x^{3}}{3125}-\frac {66 x^{2}}{15625}+\frac {834 x}{78125}-\frac {1008}{390625}\right ) \\ \end{align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= x^{3} c_{4} +c_{3} x^{2}+c_{2} x +c_{1} +{\mathrm e}^{\left (-1+2 i\right ) x} c_{5} +{\mathrm e}^{\left (-1-2 i\right ) x} c_{6} +\frac {\left (\left (183750 x -1176000\right ) \cos \left (2 x \right )+\left (-630000 x -889875\right ) \sin \left (2 x \right )[phone] x^{2}[phone] x [phone]\right ) {\mathrm e}^{-x}}{65625000}+\frac {x^{7}}{4200}-\frac {x^{6}}{1500}-\frac {x^{5}}{2500}+\frac {3 x^{4}}{625}-\frac {19 x^{3}}{3125}-\frac {66 x^{2}}{15625}+\frac {834 x}{78125}-\frac {1008}{390625} \\ \end{align*}

Verification of solutions

\[ y = x^{3} c_{4} +c_{3} x^{2}+c_{2} x +c_{1} +{\mathrm e}^{\left (-1+2 i\right ) x} c_{5} +{\mathrm e}^{\left (-1-2 i\right ) x} c_{6} +\frac {\left (\left (183750 x -1176000\right ) \cos \left (2 x \right )+\left (-630000 x -889875\right ) \sin \left (2 x \right )[phone] x^{2}[phone] x [phone]\right ) {\mathrm e}^{-x}}{65625000}+\frac {x^{7}}{4200}-\frac {x^{6}}{1500}-\frac {x^{5}}{2500}+\frac {3 x^{4}}{625}-\frac {19 x^{3}}{3125}-\frac {66 x^{2}}{15625}+\frac {834 x}{78125}-\frac {1008}{390625} \] Verified OK.

Maple trace

`Methods for high order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying high order exact linear fully integrable 
trying differential order: 6; linear nonhomogeneous with symmetry [0,1] 
-> Calling odsolve with the ODE`, diff(diff(_b(_a), _a), _a) = _a^2*exp(-_a)+_a^3+exp(-_a)*sin(2*_a)-2*(diff(_b(_a), _a))-5*_b(_a), 
   Methods for second order ODEs: 
   --- Trying classification methods --- 
   trying a quadrature 
   trying high order exact linear fully integrable 
   trying differential order: 2; linear nonhomogeneous with symmetry [0,1] 
   trying a double symmetry of the form [xi=0, eta=F(x)] 
   -> Try solving first the homogeneous part of the ODE 
      checking if the LODE has constant coefficients 
      <- constant coefficients successful 
   <- solving first the homogeneous part of the ODE successful 
<- differential order: 6; linear nonhomogeneous with symmetry [0,1] successful`
 

✓ Solution by Maple

Time used: 0.031 (sec). Leaf size: 91

dsolve(diff(y(x),x$6)+2*diff(y(x),x$5)+5*diff(y(x),x$4)=x^3+x^2*exp(-x)+exp(-x)*sin(2*x),y(x), singsol=all)
 

\[ y \left (x \right ) = c_{5} x +c_{6} +\frac {\left (\int \left (\left (\left (-330 x +1320 c_{1} +240 c_{2} +69\right ) \cos \left (2 x \right )+\left (60 x -240 c_{1} +1320 c_{2} +567\right ) \sin \left (2 x \right )-3750 x^{2}-22500 x -43125\right ) {\mathrm e}^{-x}+25 x^{6}-60 x^{5}-30 x^{4}+288 x^{3}+7500 c_{3} x^{2}+15000 c_{4} x \right )d x \right )}{15000} \]

✓ Solution by Mathematica

Time used: 11.809 (sec). Leaf size: 119

DSolve[y''''''[x]+2*y'''''[x]+5*y''''[x]==x^3+x^2*Exp[-x]+Exp[-x]*Sin[2*x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_6 x^3+c_5 x^2+\frac {e^{-x} \left (10 \left (25 e^x x^7-70 e^x x^6-42 e^x x^5+504 e^x x^4+26250 x^2+210000 x+511875\right )+84 (35 x-2 (97+240 c_1+70 c_2)) \cos (2 x)-21 (480 x+643+560 c_1-1920 c_2) \sin (2 x)\right )}{1050000}+c_4 x+c_3 \]