problem 1580

Internal problem ID [9902]
Internal ﬁle name [OUTPUT/8849_Monday_June_06_2022_05_37_50_AM_87792161/index.tex]

Book: Diﬀerential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 5, linear ﬁfth and higher order
Problem number: 1580.
ODE order: 6.
ODE degree: 1.

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

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

Therefore the homogeneous solution is \[ y_h(x)={\mathrm e}^{-\frac {\sqrt {2+2 i \sqrt {3}}\, x}{2}} c_{1} +{\mathrm e}^{i x} c_{2} +{\mathrm e}^{\frac {\sqrt {2-2 i \sqrt {3}}\, x}{2}} c_{3} +{\mathrm e}^{-i x} c_{4} +{\mathrm e}^{-\frac {\sqrt {2-2 i \sqrt {3}}\, x}{2}} c_{5} +{\mathrm e}^{\frac {\sqrt {2+2 i \sqrt {3}}\, x}{2}} c_{6} \] The fundamental set of solutions for the homogeneous solution are the following \begin{align*} y_1 &= {\mathrm e}^{-\frac {\sqrt {2+2 i \sqrt {3}}\, x}{2}} \\ y_2 &= {\mathrm e}^{i x} \\ y_3 &= {\mathrm e}^{\frac {\sqrt {2-2 i \sqrt {3}}\, x}{2}} \\ y_4 &= {\mathrm e}^{-i x} \\ y_5 &= {\mathrm e}^{-\frac {\sqrt {2-2 i \sqrt {3}}\, x}{2}} \\ y_6 &= {\mathrm e}^{\frac {\sqrt {2+2 i \sqrt {3}}\, x}{2}} \\ \end{align*} Now the particular solution to the given ODE is found \[ y^{\left (6\right )}+y = \sin \left (\frac {3 x}{2}\right ) \sin \left (\frac {x}{2}\right ) \] 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} {\mathrm e}^{-\frac {\left (\sqrt {3}+i\right ) x}{2}} & {\mathrm e}^{i x} & {\mathrm e}^{-\frac {\left (i-\sqrt {3}\right ) x}{2}} & {\mathrm e}^{-i x} & {\mathrm e}^{\frac {\left (i-\sqrt {3}\right ) x}{2}} & {\mathrm e}^{\frac {\left (\sqrt {3}+i\right ) x}{2}} \\ -\frac {\left (\sqrt {3}+i\right ) {\mathrm e}^{-\frac {\left (\sqrt {3}+i\right ) x}{2}}}{2} & i {\mathrm e}^{i x} & -\frac {\left (i-\sqrt {3}\right ) {\mathrm e}^{-\frac {\left (i-\sqrt {3}\right ) x}{2}}}{2} & -i {\mathrm e}^{-i x} & \frac {\left (i-\sqrt {3}\right ) {\mathrm e}^{\frac {\left (i-\sqrt {3}\right ) x}{2}}}{2} & \frac {\left (\sqrt {3}+i\right ) {\mathrm e}^{\frac {\left (\sqrt {3}+i\right ) x}{2}}}{2} \\ \frac {\left (1+i \sqrt {3}\right ) {\mathrm e}^{-\frac {\left (\sqrt {3}+i\right ) x}{2}}}{2} & -{\mathrm e}^{i x} & -\frac {\left (-1+i \sqrt {3}\right ) {\mathrm e}^{-\frac {\left (i-\sqrt {3}\right ) x}{2}}}{2} & -{\mathrm e}^{-i x} & -\frac {\left (-1+i \sqrt {3}\right ) {\mathrm e}^{\frac {\left (i-\sqrt {3}\right ) x}{2}}}{2} & \frac {\left (1+i \sqrt {3}\right ) {\mathrm e}^{\frac {\left (\sqrt {3}+i\right ) x}{2}}}{2} \\ -i {\mathrm e}^{-\frac {\left (\sqrt {3}+i\right ) x}{2}} & -i {\mathrm e}^{i x} & -i {\mathrm e}^{-\frac {\left (i-\sqrt {3}\right ) x}{2}} & i {\mathrm e}^{-i x} & i {\mathrm e}^{\frac {\left (i-\sqrt {3}\right ) x}{2}} & i {\mathrm e}^{\frac {\left (\sqrt {3}+i\right ) x}{2}} \\ \frac {\left (1+i \sqrt {3}\right )^{2} {\mathrm e}^{-\frac {\left (\sqrt {3}+i\right ) x}{2}}}{4} & {\mathrm e}^{i x} & \frac {\left (-1+i \sqrt {3}\right )^{2} {\mathrm e}^{-\frac {\left (i-\sqrt {3}\right ) x}{2}}}{4} & {\mathrm e}^{-i x} & \frac {\left (-1+i \sqrt {3}\right )^{2} {\mathrm e}^{\frac {\left (i-\sqrt {3}\right ) x}{2}}}{4} & \frac {\left (1+i \sqrt {3}\right )^{2} {\mathrm e}^{\frac {\left (\sqrt {3}+i\right ) x}{2}}}{4} \\ -\frac {\left (i-\sqrt {3}\right ) {\mathrm e}^{-\frac {\left (\sqrt {3}+i\right ) x}{2}}}{2} & i {\mathrm e}^{i x} & -\frac {\left (\sqrt {3}+i\right ) {\mathrm e}^{-\frac {\left (i-\sqrt {3}\right ) x}{2}}}{2} & -i {\mathrm e}^{-i x} & \frac {\left (\sqrt {3}+i\right ) {\mathrm e}^{\frac {\left (i-\sqrt {3}\right ) x}{2}}}{2} & \frac {\left (i-\sqrt {3}\right ) {\mathrm e}^{\frac {\left (\sqrt {3}+i\right ) x}{2}}}{2} \end {array}\right ] \\ |W| &= 216 i {\mathrm e}^{-\frac {\left (\sqrt {3}+i\right ) x}{2}} {\mathrm e}^{\frac {\left (i-\sqrt {3}\right ) x}{2}} {\mathrm e}^{i x} {\mathrm e}^{-i x} {\mathrm e}^{\frac {\left (\sqrt {3}+i\right ) x}{2}} {\mathrm e}^{-\frac {\left (i-\sqrt {3}\right ) x}{2}} \end {align*}

Now we determine \(W_i\) for each \(U_i\). \begin {align*} W_1(x) &= \det \,\left [\begin {array}{ccccc} {\mathrm e}^{i x} & {\mathrm e}^{-\frac {\left (i-\sqrt {3}\right ) x}{2}} & {\mathrm e}^{-i x} & {\mathrm e}^{\frac {\left (i-\sqrt {3}\right ) x}{2}} & {\mathrm e}^{\frac {\left (\sqrt {3}+i\right ) x}{2}} \\ i {\mathrm e}^{i x} & -\frac {\left (i-\sqrt {3}\right ) {\mathrm e}^{-\frac {\left (i-\sqrt {3}\right ) x}{2}}}{2} & -i {\mathrm e}^{-i x} & \frac {\left (i-\sqrt {3}\right ) {\mathrm e}^{\frac {\left (i-\sqrt {3}\right ) x}{2}}}{2} & \frac {\left (\sqrt {3}+i\right ) {\mathrm e}^{\frac {\left (\sqrt {3}+i\right ) x}{2}}}{2} \\ -{\mathrm e}^{i x} & -\frac {\left (-1+i \sqrt {3}\right ) {\mathrm e}^{-\frac {\left (i-\sqrt {3}\right ) x}{2}}}{2} & -{\mathrm e}^{-i x} & -\frac {\left (-1+i \sqrt {3}\right ) {\mathrm e}^{\frac {\left (i-\sqrt {3}\right ) x}{2}}}{2} & \frac {\left (1+i \sqrt {3}\right ) {\mathrm e}^{\frac {\left (\sqrt {3}+i\right ) x}{2}}}{2} \\ -i {\mathrm e}^{i x} & -i {\mathrm e}^{-\frac {\left (i-\sqrt {3}\right ) x}{2}} & i {\mathrm e}^{-i x} & i {\mathrm e}^{\frac {\left (i-\sqrt {3}\right ) x}{2}} & i {\mathrm e}^{\frac {\left (\sqrt {3}+i\right ) x}{2}} \\ {\mathrm e}^{i x} & \frac {\left (-1+i \sqrt {3}\right )^{2} {\mathrm e}^{-\frac {\left (i-\sqrt {3}\right ) x}{2}}}{4} & {\mathrm e}^{-i x} & \frac {\left (-1+i \sqrt {3}\right )^{2} {\mathrm e}^{\frac {\left (i-\sqrt {3}\right ) x}{2}}}{4} & \frac {\left (1+i \sqrt {3}\right )^{2} {\mathrm e}^{\frac {\left (\sqrt {3}+i\right ) x}{2}}}{4} \end {array}\right ] \\ &= -18 \,{\mathrm e}^{\frac {\left (\sqrt {3}+i\right ) x}{2}} \left (-1+i \sqrt {3}\right ) \end {align*}

\begin {align*} W_2(x) &= \det \,\left [\begin {array}{ccccc} {\mathrm e}^{-\frac {\left (\sqrt {3}+i\right ) x}{2}} & {\mathrm e}^{-\frac {\left (i-\sqrt {3}\right ) x}{2}} & {\mathrm e}^{-i x} & {\mathrm e}^{\frac {\left (i-\sqrt {3}\right ) x}{2}} & {\mathrm e}^{\frac {\left (\sqrt {3}+i\right ) x}{2}} \\ -\frac {\left (\sqrt {3}+i\right ) {\mathrm e}^{-\frac {\left (\sqrt {3}+i\right ) x}{2}}}{2} & -\frac {\left (i-\sqrt {3}\right ) {\mathrm e}^{-\frac {\left (i-\sqrt {3}\right ) x}{2}}}{2} & -i {\mathrm e}^{-i x} & \frac {\left (i-\sqrt {3}\right ) {\mathrm e}^{\frac {\left (i-\sqrt {3}\right ) x}{2}}}{2} & \frac {\left (\sqrt {3}+i\right ) {\mathrm e}^{\frac {\left (\sqrt {3}+i\right ) x}{2}}}{2} \\ \frac {\left (1+i \sqrt {3}\right ) {\mathrm e}^{-\frac {\left (\sqrt {3}+i\right ) x}{2}}}{2} & -\frac {\left (-1+i \sqrt {3}\right ) {\mathrm e}^{-\frac {\left (i-\sqrt {3}\right ) x}{2}}}{2} & -{\mathrm e}^{-i x} & -\frac {\left (-1+i \sqrt {3}\right ) {\mathrm e}^{\frac {\left (i-\sqrt {3}\right ) x}{2}}}{2} & \frac {\left (1+i \sqrt {3}\right ) {\mathrm e}^{\frac {\left (\sqrt {3}+i\right ) x}{2}}}{2} \\ -i {\mathrm e}^{-\frac {\left (\sqrt {3}+i\right ) x}{2}} & -i {\mathrm e}^{-\frac {\left (i-\sqrt {3}\right ) x}{2}} & i {\mathrm e}^{-i x} & i {\mathrm e}^{\frac {\left (i-\sqrt {3}\right ) x}{2}} & i {\mathrm e}^{\frac {\left (\sqrt {3}+i\right ) x}{2}} \\ \frac {\left (1+i \sqrt {3}\right )^{2} {\mathrm e}^{-\frac {\left (\sqrt {3}+i\right ) x}{2}}}{4} & \frac {\left (-1+i \sqrt {3}\right )^{2} {\mathrm e}^{-\frac {\left (i-\sqrt {3}\right ) x}{2}}}{4} & {\mathrm e}^{-i x} & \frac {\left (-1+i \sqrt {3}\right )^{2} {\mathrm e}^{\frac {\left (i-\sqrt {3}\right ) x}{2}}}{4} & \frac {\left (1+i \sqrt {3}\right )^{2} {\mathrm e}^{\frac {\left (\sqrt {3}+i\right ) x}{2}}}{4} \end {array}\right ] \\ &= 36 \,{\mathrm e}^{-i x} \end {align*}

\begin {align*} W_3(x) &= \det \,\left [\begin {array}{ccccc} {\mathrm e}^{-\frac {\left (\sqrt {3}+i\right ) x}{2}} & {\mathrm e}^{i x} & {\mathrm e}^{-i x} & {\mathrm e}^{\frac {\left (i-\sqrt {3}\right ) x}{2}} & {\mathrm e}^{\frac {\left (\sqrt {3}+i\right ) x}{2}} \\ -\frac {\left (\sqrt {3}+i\right ) {\mathrm e}^{-\frac {\left (\sqrt {3}+i\right ) x}{2}}}{2} & i {\mathrm e}^{i x} & -i {\mathrm e}^{-i x} & \frac {\left (i-\sqrt {3}\right ) {\mathrm e}^{\frac {\left (i-\sqrt {3}\right ) x}{2}}}{2} & \frac {\left (\sqrt {3}+i\right ) {\mathrm e}^{\frac {\left (\sqrt {3}+i\right ) x}{2}}}{2} \\ \frac {\left (1+i \sqrt {3}\right ) {\mathrm e}^{-\frac {\left (\sqrt {3}+i\right ) x}{2}}}{2} & -{\mathrm e}^{i x} & -{\mathrm e}^{-i x} & -\frac {\left (-1+i \sqrt {3}\right ) {\mathrm e}^{\frac {\left (i-\sqrt {3}\right ) x}{2}}}{2} & \frac {\left (1+i \sqrt {3}\right ) {\mathrm e}^{\frac {\left (\sqrt {3}+i\right ) x}{2}}}{2} \\ -i {\mathrm e}^{-\frac {\left (\sqrt {3}+i\right ) x}{2}} & -i {\mathrm e}^{i x} & i {\mathrm e}^{-i x} & i {\mathrm e}^{\frac {\left (i-\sqrt {3}\right ) x}{2}} & i {\mathrm e}^{\frac {\left (\sqrt {3}+i\right ) x}{2}} \\ \frac {\left (1+i \sqrt {3}\right )^{2} {\mathrm e}^{-\frac {\left (\sqrt {3}+i\right ) x}{2}}}{4} & {\mathrm e}^{i x} & {\mathrm e}^{-i x} & \frac {\left (-1+i \sqrt {3}\right )^{2} {\mathrm e}^{\frac {\left (i-\sqrt {3}\right ) x}{2}}}{4} & \frac {\left (1+i \sqrt {3}\right )^{2} {\mathrm e}^{\frac {\left (\sqrt {3}+i\right ) x}{2}}}{4} \end {array}\right ] \\ &= 18 \,{\mathrm e}^{-\frac {x \left (-i+\sqrt {3}\right )}{2}} \left (1+i \sqrt {3}\right ) \end {align*}

\begin {align*} W_4(x) &= \det \,\left [\begin {array}{ccccc} {\mathrm e}^{-\frac {\left (\sqrt {3}+i\right ) x}{2}} & {\mathrm e}^{i x} & {\mathrm e}^{-\frac {\left (i-\sqrt {3}\right ) x}{2}} & {\mathrm e}^{\frac {\left (i-\sqrt {3}\right ) x}{2}} & {\mathrm e}^{\frac {\left (\sqrt {3}+i\right ) x}{2}} \\ -\frac {\left (\sqrt {3}+i\right ) {\mathrm e}^{-\frac {\left (\sqrt {3}+i\right ) x}{2}}}{2} & i {\mathrm e}^{i x} & -\frac {\left (i-\sqrt {3}\right ) {\mathrm e}^{-\frac {\left (i-\sqrt {3}\right ) x}{2}}}{2} & \frac {\left (i-\sqrt {3}\right ) {\mathrm e}^{\frac {\left (i-\sqrt {3}\right ) x}{2}}}{2} & \frac {\left (\sqrt {3}+i\right ) {\mathrm e}^{\frac {\left (\sqrt {3}+i\right ) x}{2}}}{2} \\ \frac {\left (1+i \sqrt {3}\right ) {\mathrm e}^{-\frac {\left (\sqrt {3}+i\right ) x}{2}}}{2} & -{\mathrm e}^{i x} & -\frac {\left (-1+i \sqrt {3}\right ) {\mathrm e}^{-\frac {\left (i-\sqrt {3}\right ) x}{2}}}{2} & -\frac {\left (-1+i \sqrt {3}\right ) {\mathrm e}^{\frac {\left (i-\sqrt {3}\right ) x}{2}}}{2} & \frac {\left (1+i \sqrt {3}\right ) {\mathrm e}^{\frac {\left (\sqrt {3}+i\right ) x}{2}}}{2} \\ -i {\mathrm e}^{-\frac {\left (\sqrt {3}+i\right ) x}{2}} & -i {\mathrm e}^{i x} & -i {\mathrm e}^{-\frac {\left (i-\sqrt {3}\right ) x}{2}} & i {\mathrm e}^{\frac {\left (i-\sqrt {3}\right ) x}{2}} & i {\mathrm e}^{\frac {\left (\sqrt {3}+i\right ) x}{2}} \\ \frac {\left (1+i \sqrt {3}\right )^{2} {\mathrm e}^{-\frac {\left (\sqrt {3}+i\right ) x}{2}}}{4} & {\mathrm e}^{i x} & \frac {\left (-1+i \sqrt {3}\right )^{2} {\mathrm e}^{-\frac {\left (i-\sqrt {3}\right ) x}{2}}}{4} & \frac {\left (-1+i \sqrt {3}\right )^{2} {\mathrm e}^{\frac {\left (i-\sqrt {3}\right ) x}{2}}}{4} & \frac {\left (1+i \sqrt {3}\right )^{2} {\mathrm e}^{\frac {\left (\sqrt {3}+i\right ) x}{2}}}{4} \end {array}\right ] \\ &= -36 \,{\mathrm e}^{i x} \end {align*}

\begin {align*} W_5(x) &= \det \,\left [\begin {array}{ccccc} {\mathrm e}^{-\frac {\left (\sqrt {3}+i\right ) x}{2}} & {\mathrm e}^{i x} & {\mathrm e}^{-\frac {\left (i-\sqrt {3}\right ) x}{2}} & {\mathrm e}^{-i x} & {\mathrm e}^{\frac {\left (\sqrt {3}+i\right ) x}{2}} \\ -\frac {\left (\sqrt {3}+i\right ) {\mathrm e}^{-\frac {\left (\sqrt {3}+i\right ) x}{2}}}{2} & i {\mathrm e}^{i x} & -\frac {\left (i-\sqrt {3}\right ) {\mathrm e}^{-\frac {\left (i-\sqrt {3}\right ) x}{2}}}{2} & -i {\mathrm e}^{-i x} & \frac {\left (\sqrt {3}+i\right ) {\mathrm e}^{\frac {\left (\sqrt {3}+i\right ) x}{2}}}{2} \\ \frac {\left (1+i \sqrt {3}\right ) {\mathrm e}^{-\frac {\left (\sqrt {3}+i\right ) x}{2}}}{2} & -{\mathrm e}^{i x} & -\frac {\left (-1+i \sqrt {3}\right ) {\mathrm e}^{-\frac {\left (i-\sqrt {3}\right ) x}{2}}}{2} & -{\mathrm e}^{-i x} & \frac {\left (1+i \sqrt {3}\right ) {\mathrm e}^{\frac {\left (\sqrt {3}+i\right ) x}{2}}}{2} \\ -i {\mathrm e}^{-\frac {\left (\sqrt {3}+i\right ) x}{2}} & -i {\mathrm e}^{i x} & -i {\mathrm e}^{-\frac {\left (i-\sqrt {3}\right ) x}{2}} & i {\mathrm e}^{-i x} & i {\mathrm e}^{\frac {\left (\sqrt {3}+i\right ) x}{2}} \\ \frac {\left (1+i \sqrt {3}\right )^{2} {\mathrm e}^{-\frac {\left (\sqrt {3}+i\right ) x}{2}}}{4} & {\mathrm e}^{i x} & \frac {\left (-1+i \sqrt {3}\right )^{2} {\mathrm e}^{-\frac {\left (i-\sqrt {3}\right ) x}{2}}}{4} & {\mathrm e}^{-i x} & \frac {\left (1+i \sqrt {3}\right )^{2} {\mathrm e}^{\frac {\left (\sqrt {3}+i\right ) x}{2}}}{4} \end {array}\right ] \\ &= -18 \,{\mathrm e}^{\frac {x \left (-i+\sqrt {3}\right )}{2}} \left (1+i \sqrt {3}\right ) \end {align*}

\begin {align*} W_6(x) &= \det \,\left [\begin {array}{ccccc} {\mathrm e}^{-\frac {\left (\sqrt {3}+i\right ) x}{2}} & {\mathrm e}^{i x} & {\mathrm e}^{-\frac {\left (i-\sqrt {3}\right ) x}{2}} & {\mathrm e}^{-i x} & {\mathrm e}^{\frac {\left (i-\sqrt {3}\right ) x}{2}} \\ -\frac {\left (\sqrt {3}+i\right ) {\mathrm e}^{-\frac {\left (\sqrt {3}+i\right ) x}{2}}}{2} & i {\mathrm e}^{i x} & -\frac {\left (i-\sqrt {3}\right ) {\mathrm e}^{-\frac {\left (i-\sqrt {3}\right ) x}{2}}}{2} & -i {\mathrm e}^{-i x} & \frac {\left (i-\sqrt {3}\right ) {\mathrm e}^{\frac {\left (i-\sqrt {3}\right ) x}{2}}}{2} \\ \frac {\left (1+i \sqrt {3}\right ) {\mathrm e}^{-\frac {\left (\sqrt {3}+i\right ) x}{2}}}{2} & -{\mathrm e}^{i x} & -\frac {\left (-1+i \sqrt {3}\right ) {\mathrm e}^{-\frac {\left (i-\sqrt {3}\right ) x}{2}}}{2} & -{\mathrm e}^{-i x} & -\frac {\left (-1+i \sqrt {3}\right ) {\mathrm e}^{\frac {\left (i-\sqrt {3}\right ) x}{2}}}{2} \\ -i {\mathrm e}^{-\frac {\left (\sqrt {3}+i\right ) x}{2}} & -i {\mathrm e}^{i x} & -i {\mathrm e}^{-\frac {\left (i-\sqrt {3}\right ) x}{2}} & i {\mathrm e}^{-i x} & i {\mathrm e}^{\frac {\left (i-\sqrt {3}\right ) x}{2}} \\ \frac {\left (1+i \sqrt {3}\right )^{2} {\mathrm e}^{-\frac {\left (\sqrt {3}+i\right ) x}{2}}}{4} & {\mathrm e}^{i x} & \frac {\left (-1+i \sqrt {3}\right )^{2} {\mathrm e}^{-\frac {\left (i-\sqrt {3}\right ) x}{2}}}{4} & {\mathrm e}^{-i x} & \frac {\left (-1+i \sqrt {3}\right )^{2} {\mathrm e}^{\frac {\left (i-\sqrt {3}\right ) x}{2}}}{4} \end {array}\right ] \\ &= -18 \,{\mathrm e}^{-\frac {\left (\sqrt {3}+i\right ) x}{2}} \left (-1+i \sqrt {3}\right ) \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 (\sin \left (\frac {3 x}{2}\right ) \sin \left (\frac {x}{2}\right )\right ) \left (-18 \,{\mathrm e}^{\frac {\left (\sqrt {3}+i\right ) x}{2}} \left (-1+i \sqrt {3}\right )\right )}{\left (1\right ) \left (216 i\right )} \, dx} \\ &= - \int { \frac {-18 \sin \left (\frac {3 x}{2}\right ) \sin \left (\frac {x}{2}\right ) {\mathrm e}^{\frac {\left (\sqrt {3}+i\right ) x}{2}} \left (-1+i \sqrt {3}\right )}{216 i} \, dx}\\ &= - \int {\left (-\frac {\sin \left (\frac {3 x}{2}\right ) \sin \left (\frac {x}{2}\right ) {\mathrm e}^{\frac {\left (\sqrt {3}+i\right ) x}{2}} \left (\sqrt {3}+i\right )}{12}\right ) \, dx} \\ &= -\left (-\frac {\sqrt {3}}{12}-\frac {i}{12}\right ) \left (\frac {\left (\frac {i}{2}+\frac {\sqrt {3}}{2}\right ) {\mathrm e}^{\left (\frac {i}{2}+\frac {\sqrt {3}}{2}\right ) x} \cos \left (x \right )}{2 \left (\frac {i}{2}+\frac {\sqrt {3}}{2}\right )^{2}+2}+\frac {{\mathrm e}^{\left (\frac {i}{2}+\frac {\sqrt {3}}{2}\right ) x} \sin \left (x \right )}{2 \left (\frac {i}{2}+\frac {\sqrt {3}}{2}\right )^{2}+2}-\frac {\left (\frac {i}{2}+\frac {\sqrt {3}}{2}\right ) {\mathrm e}^{\left (\frac {i}{2}+\frac {\sqrt {3}}{2}\right ) x} \cos \left (2 x \right )}{2 \left (\left (\frac {i}{2}+\frac {\sqrt {3}}{2}\right )^{2}+4\right )}-\frac {{\mathrm e}^{\left (\frac {i}{2}+\frac {\sqrt {3}}{2}\right ) x} \sin \left (2 x \right )}{\left (\frac {i}{2}+\frac {\sqrt {3}}{2}\right )^{2}+4}\right ) \\ &= -\left (-\frac {\sqrt {3}}{12}-\frac {i}{12}\right ) \left (\frac {\left (\frac {i}{2}+\frac {\sqrt {3}}{2}\right ) {\mathrm e}^{\left (\frac {i}{2}+\frac {\sqrt {3}}{2}\right ) x} \cos \left (x \right )}{2 \left (\frac {i}{2}+\frac {\sqrt {3}}{2}\right )^{2}+2}+\frac {{\mathrm e}^{\left (\frac {i}{2}+\frac {\sqrt {3}}{2}\right ) x} \sin \left (x \right )}{2 \left (\frac {i}{2}+\frac {\sqrt {3}}{2}\right )^{2}+2}-\frac {\left (\frac {i}{2}+\frac {\sqrt {3}}{2}\right ) {\mathrm e}^{\left (\frac {i}{2}+\frac {\sqrt {3}}{2}\right ) x} \cos \left (2 x \right )}{2 \left (\left (\frac {i}{2}+\frac {\sqrt {3}}{2}\right )^{2}+4\right )}-\frac {{\mathrm e}^{\left (\frac {i}{2}+\frac {\sqrt {3}}{2}\right ) x} \sin \left (2 x \right )}{\left (\frac {i}{2}+\frac {\sqrt {3}}{2}\right )^{2}+4}\right ) \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 (\sin \left (\frac {3 x}{2}\right ) \sin \left (\frac {x}{2}\right )\right ) \left (36 \,{\mathrm e}^{-i x}\right )}{\left (1\right ) \left (216 i\right )} \, dx} \\ &= \int { \frac {36 \sin \left (\frac {3 x}{2}\right ) \sin \left (\frac {x}{2}\right ) {\mathrm e}^{-i x}}{216 i} \, dx}\\ &= \int {\left (-\frac {i \sin \left (\frac {3 x}{2}\right ) \sin \left (\frac {x}{2}\right ) {\mathrm e}^{-i x}}{6}\right ) \, dx}\\ &= \int -\frac {i \sin \left (\frac {3 x}{2}\right ) \sin \left (\frac {x}{2}\right ) {\mathrm e}^{-i x}}{6}d x \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 (\sin \left (\frac {3 x}{2}\right ) \sin \left (\frac {x}{2}\right )\right ) \left (18 \,{\mathrm e}^{-\frac {x \left (-i+\sqrt {3}\right )}{2}} \left (1+i \sqrt {3}\right )\right )}{\left (1\right ) \left (216 i\right )} \, dx} \\ &= - \int { \frac {18 \sin \left (\frac {3 x}{2}\right ) \sin \left (\frac {x}{2}\right ) {\mathrm e}^{-\frac {x \left (-i+\sqrt {3}\right )}{2}} \left (1+i \sqrt {3}\right )}{216 i} \, dx}\\ &= - \int {\left (-\frac {\sin \left (\frac {3 x}{2}\right ) \sin \left (\frac {x}{2}\right ) {\mathrm e}^{\frac {\left (i-\sqrt {3}\right ) x}{2}} \left (i-\sqrt {3}\right )}{12}\right ) \, dx} \\ &= -\left (-\frac {i}{12}+\frac {\sqrt {3}}{12}\right ) \left (\frac {\left (\frac {i}{2}-\frac {\sqrt {3}}{2}\right ) {\mathrm e}^{\left (\frac {i}{2}-\frac {\sqrt {3}}{2}\right ) x} \cos \left (x \right )}{2 \left (\frac {i}{2}-\frac {\sqrt {3}}{2}\right )^{2}+2}+\frac {{\mathrm e}^{\left (\frac {i}{2}-\frac {\sqrt {3}}{2}\right ) x} \sin \left (x \right )}{2 \left (\frac {i}{2}-\frac {\sqrt {3}}{2}\right )^{2}+2}-\frac {\left (\frac {i}{2}-\frac {\sqrt {3}}{2}\right ) {\mathrm e}^{\left (\frac {i}{2}-\frac {\sqrt {3}}{2}\right ) x} \cos \left (2 x \right )}{2 \left (\left (\frac {i}{2}-\frac {\sqrt {3}}{2}\right )^{2}+4\right )}-\frac {{\mathrm e}^{\left (\frac {i}{2}-\frac {\sqrt {3}}{2}\right ) x} \sin \left (2 x \right )}{\left (\frac {i}{2}-\frac {\sqrt {3}}{2}\right )^{2}+4}\right ) \\ &= -\left (-\frac {i}{12}+\frac {\sqrt {3}}{12}\right ) \left (\frac {\left (\frac {i}{2}-\frac {\sqrt {3}}{2}\right ) {\mathrm e}^{\left (\frac {i}{2}-\frac {\sqrt {3}}{2}\right ) x} \cos \left (x \right )}{2 \left (\frac {i}{2}-\frac {\sqrt {3}}{2}\right )^{2}+2}+\frac {{\mathrm e}^{\left (\frac {i}{2}-\frac {\sqrt {3}}{2}\right ) x} \sin \left (x \right )}{2 \left (\frac {i}{2}-\frac {\sqrt {3}}{2}\right )^{2}+2}-\frac {\left (\frac {i}{2}-\frac {\sqrt {3}}{2}\right ) {\mathrm e}^{\left (\frac {i}{2}-\frac {\sqrt {3}}{2}\right ) x} \cos \left (2 x \right )}{2 \left (\left (\frac {i}{2}-\frac {\sqrt {3}}{2}\right )^{2}+4\right )}-\frac {{\mathrm e}^{\left (\frac {i}{2}-\frac {\sqrt {3}}{2}\right ) x} \sin \left (2 x \right )}{\left (\frac {i}{2}-\frac {\sqrt {3}}{2}\right )^{2}+4}\right ) \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 (\sin \left (\frac {3 x}{2}\right ) \sin \left (\frac {x}{2}\right )\right ) \left (-36 \,{\mathrm e}^{i x}\right )}{\left (1\right ) \left (216 i\right )} \, dx} \\ &= \int { \frac {-36 \sin \left (\frac {3 x}{2}\right ) \sin \left (\frac {x}{2}\right ) {\mathrm e}^{i x}}{216 i} \, dx}\\ &= \int {\left (\frac {i \sin \left (\frac {3 x}{2}\right ) \sin \left (\frac {x}{2}\right ) {\mathrm e}^{i x}}{6}\right ) \, dx}\\ &= \int \frac {i \sin \left (\frac {3 x}{2}\right ) \sin \left (\frac {x}{2}\right ) {\mathrm e}^{i x}}{6}d x \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 (\sin \left (\frac {3 x}{2}\right ) \sin \left (\frac {x}{2}\right )\right ) \left (-18 \,{\mathrm e}^{\frac {x \left (-i+\sqrt {3}\right )}{2}} \left (1+i \sqrt {3}\right )\right )}{\left (1\right ) \left (216 i\right )} \, dx} \\ &= - \int { \frac {-18 \sin \left (\frac {3 x}{2}\right ) \sin \left (\frac {x}{2}\right ) {\mathrm e}^{\frac {x \left (-i+\sqrt {3}\right )}{2}} \left (1+i \sqrt {3}\right )}{216 i} \, dx}\\ &= - \int {\left (\frac {\left (i-\sqrt {3}\right ) \sin \left (\frac {3 x}{2}\right ) \sin \left (\frac {x}{2}\right ) {\mathrm e}^{-\frac {\left (i-\sqrt {3}\right ) x}{2}}}{12}\right ) \, dx} \\ &= -\left (\frac {i}{12}-\frac {\sqrt {3}}{12}\right ) \left (\frac {\left (-\frac {i}{2}+\frac {\sqrt {3}}{2}\right ) {\mathrm e}^{\left (-\frac {i}{2}+\frac {\sqrt {3}}{2}\right ) x} \cos \left (x \right )}{2 \left (-\frac {i}{2}+\frac {\sqrt {3}}{2}\right )^{2}+2}+\frac {{\mathrm e}^{\left (-\frac {i}{2}+\frac {\sqrt {3}}{2}\right ) x} \sin \left (x \right )}{2 \left (-\frac {i}{2}+\frac {\sqrt {3}}{2}\right )^{2}+2}-\frac {\left (-\frac {i}{2}+\frac {\sqrt {3}}{2}\right ) {\mathrm e}^{\left (-\frac {i}{2}+\frac {\sqrt {3}}{2}\right ) x} \cos \left (2 x \right )}{2 \left (\left (-\frac {i}{2}+\frac {\sqrt {3}}{2}\right )^{2}+4\right )}-\frac {{\mathrm e}^{\left (-\frac {i}{2}+\frac {\sqrt {3}}{2}\right ) x} \sin \left (2 x \right )}{\left (-\frac {i}{2}+\frac {\sqrt {3}}{2}\right )^{2}+4}\right ) \\ &= -\left (\frac {i}{12}-\frac {\sqrt {3}}{12}\right ) \left (\frac {\left (-\frac {i}{2}+\frac {\sqrt {3}}{2}\right ) {\mathrm e}^{\left (-\frac {i}{2}+\frac {\sqrt {3}}{2}\right ) x} \cos \left (x \right )}{2 \left (-\frac {i}{2}+\frac {\sqrt {3}}{2}\right )^{2}+2}+\frac {{\mathrm e}^{\left (-\frac {i}{2}+\frac {\sqrt {3}}{2}\right ) x} \sin \left (x \right )}{2 \left (-\frac {i}{2}+\frac {\sqrt {3}}{2}\right )^{2}+2}-\frac {\left (-\frac {i}{2}+\frac {\sqrt {3}}{2}\right ) {\mathrm e}^{\left (-\frac {i}{2}+\frac {\sqrt {3}}{2}\right ) x} \cos \left (2 x \right )}{2 \left (\left (-\frac {i}{2}+\frac {\sqrt {3}}{2}\right )^{2}+4\right )}-\frac {{\mathrm e}^{\left (-\frac {i}{2}+\frac {\sqrt {3}}{2}\right ) x} \sin \left (2 x \right )}{\left (-\frac {i}{2}+\frac {\sqrt {3}}{2}\right )^{2}+4}\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 (\sin \left (\frac {3 x}{2}\right ) \sin \left (\frac {x}{2}\right )\right ) \left (-18 \,{\mathrm e}^{-\frac {\left (\sqrt {3}+i\right ) x}{2}} \left (-1+i \sqrt {3}\right )\right )}{\left (1\right ) \left (216 i\right )} \, dx} \\ &= \int { \frac {-18 \sin \left (\frac {3 x}{2}\right ) \sin \left (\frac {x}{2}\right ) {\mathrm e}^{-\frac {\left (\sqrt {3}+i\right ) x}{2}} \left (-1+i \sqrt {3}\right )}{216 i} \, dx}\\ &= \int {\left (-\frac {\sin \left (\frac {3 x}{2}\right ) \sin \left (\frac {x}{2}\right ) {\mathrm e}^{-\frac {\left (\sqrt {3}+i\right ) x}{2}} \left (\sqrt {3}+i\right )}{12}\right ) \, dx} \\ &= \left (-\frac {\sqrt {3}}{12}-\frac {i}{12}\right ) \left (\frac {\left (-\frac {i}{2}-\frac {\sqrt {3}}{2}\right ) {\mathrm e}^{\left (-\frac {i}{2}-\frac {\sqrt {3}}{2}\right ) x} \cos \left (x \right )}{2 \left (-\frac {i}{2}-\frac {\sqrt {3}}{2}\right )^{2}+2}+\frac {{\mathrm e}^{\left (-\frac {i}{2}-\frac {\sqrt {3}}{2}\right ) x} \sin \left (x \right )}{2 \left (-\frac {i}{2}-\frac {\sqrt {3}}{2}\right )^{2}+2}-\frac {\left (-\frac {i}{2}-\frac {\sqrt {3}}{2}\right ) {\mathrm e}^{\left (-\frac {i}{2}-\frac {\sqrt {3}}{2}\right ) x} \cos \left (2 x \right )}{2 \left (\left (-\frac {i}{2}-\frac {\sqrt {3}}{2}\right )^{2}+4\right )}-\frac {{\mathrm e}^{\left (-\frac {i}{2}-\frac {\sqrt {3}}{2}\right ) x} \sin \left (2 x \right )}{\left (-\frac {i}{2}-\frac {\sqrt {3}}{2}\right )^{2}+4}\right ) \\ &= \left (-\frac {\sqrt {3}}{12}-\frac {i}{12}\right ) \left (\frac {\left (-\frac {i}{2}-\frac {\sqrt {3}}{2}\right ) {\mathrm e}^{\left (-\frac {i}{2}-\frac {\sqrt {3}}{2}\right ) x} \cos \left (x \right )}{2 \left (-\frac {i}{2}-\frac {\sqrt {3}}{2}\right )^{2}+2}+\frac {{\mathrm e}^{\left (-\frac {i}{2}-\frac {\sqrt {3}}{2}\right ) x} \sin \left (x \right )}{2 \left (-\frac {i}{2}-\frac {\sqrt {3}}{2}\right )^{2}+2}-\frac {\left (-\frac {i}{2}-\frac {\sqrt {3}}{2}\right ) {\mathrm e}^{\left (-\frac {i}{2}-\frac {\sqrt {3}}{2}\right ) x} \cos \left (2 x \right )}{2 \left (\left (-\frac {i}{2}-\frac {\sqrt {3}}{2}\right )^{2}+4\right )}-\frac {{\mathrm e}^{\left (-\frac {i}{2}-\frac {\sqrt {3}}{2}\right ) x} \sin \left (2 x \right )}{\left (-\frac {i}{2}-\frac {\sqrt {3}}{2}\right )^{2}+4}\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 (-\left (-\frac {\sqrt {3}}{12}-\frac {i}{12}\right ) \left (\frac {\left (\frac {i}{2}+\frac {\sqrt {3}}{2}\right ) {\mathrm e}^{\left (\frac {i}{2}+\frac {\sqrt {3}}{2}\right ) x} \cos \left (x \right )}{2 \left (\frac {i}{2}+\frac {\sqrt {3}}{2}\right )^{2}+2}+\frac {{\mathrm e}^{\left (\frac {i}{2}+\frac {\sqrt {3}}{2}\right ) x} \sin \left (x \right )}{2 \left (\frac {i}{2}+\frac {\sqrt {3}}{2}\right )^{2}+2}-\frac {\left (\frac {i}{2}+\frac {\sqrt {3}}{2}\right ) {\mathrm e}^{\left (\frac {i}{2}+\frac {\sqrt {3}}{2}\right ) x} \cos \left (2 x \right )}{2 \left (\left (\frac {i}{2}+\frac {\sqrt {3}}{2}\right )^{2}+4\right )}-\frac {{\mathrm e}^{\left (\frac {i}{2}+\frac {\sqrt {3}}{2}\right ) x} \sin \left (2 x \right )}{\left (\frac {i}{2}+\frac {\sqrt {3}}{2}\right )^{2}+4}\right )\right ) \left ({\mathrm e}^{-\frac {\sqrt {2+2 i \sqrt {3}}\, x}{2}}\right ) \\ &+\left (\int -\frac {i \sin \left (\frac {3 x}{2}\right ) \sin \left (\frac {x}{2}\right ) {\mathrm e}^{-i x}}{6}d x\right ) \left ({\mathrm e}^{i x}\right ) \\ &+\left (-\left (-\frac {i}{12}+\frac {\sqrt {3}}{12}\right ) \left (\frac {\left (\frac {i}{2}-\frac {\sqrt {3}}{2}\right ) {\mathrm e}^{\left (\frac {i}{2}-\frac {\sqrt {3}}{2}\right ) x} \cos \left (x \right )}{2 \left (\frac {i}{2}-\frac {\sqrt {3}}{2}\right )^{2}+2}+\frac {{\mathrm e}^{\left (\frac {i}{2}-\frac {\sqrt {3}}{2}\right ) x} \sin \left (x \right )}{2 \left (\frac {i}{2}-\frac {\sqrt {3}}{2}\right )^{2}+2}-\frac {\left (\frac {i}{2}-\frac {\sqrt {3}}{2}\right ) {\mathrm e}^{\left (\frac {i}{2}-\frac {\sqrt {3}}{2}\right ) x} \cos \left (2 x \right )}{2 \left (\left (\frac {i}{2}-\frac {\sqrt {3}}{2}\right )^{2}+4\right )}-\frac {{\mathrm e}^{\left (\frac {i}{2}-\frac {\sqrt {3}}{2}\right ) x} \sin \left (2 x \right )}{\left (\frac {i}{2}-\frac {\sqrt {3}}{2}\right )^{2}+4}\right )\right ) \left ({\mathrm e}^{\frac {\sqrt {2-2 i \sqrt {3}}\, x}{2}}\right ) \\ &+\left (\int \frac {i \sin \left (\frac {3 x}{2}\right ) \sin \left (\frac {x}{2}\right ) {\mathrm e}^{i x}}{6}d x\right ) \left ({\mathrm e}^{-i x}\right ) \\ &+\left (-\left (\frac {i}{12}-\frac {\sqrt {3}}{12}\right ) \left (\frac {\left (-\frac {i}{2}+\frac {\sqrt {3}}{2}\right ) {\mathrm e}^{\left (-\frac {i}{2}+\frac {\sqrt {3}}{2}\right ) x} \cos \left (x \right )}{2 \left (-\frac {i}{2}+\frac {\sqrt {3}}{2}\right )^{2}+2}+\frac {{\mathrm e}^{\left (-\frac {i}{2}+\frac {\sqrt {3}}{2}\right ) x} \sin \left (x \right )}{2 \left (-\frac {i}{2}+\frac {\sqrt {3}}{2}\right )^{2}+2}-\frac {\left (-\frac {i}{2}+\frac {\sqrt {3}}{2}\right ) {\mathrm e}^{\left (-\frac {i}{2}+\frac {\sqrt {3}}{2}\right ) x} \cos \left (2 x \right )}{2 \left (\left (-\frac {i}{2}+\frac {\sqrt {3}}{2}\right )^{2}+4\right )}-\frac {{\mathrm e}^{\left (-\frac {i}{2}+\frac {\sqrt {3}}{2}\right ) x} \sin \left (2 x \right )}{\left (-\frac {i}{2}+\frac {\sqrt {3}}{2}\right )^{2}+4}\right )\right ) \left ({\mathrm e}^{-\frac {\sqrt {2-2 i \sqrt {3}}\, x}{2}}\right ) \\ &+\left (\left (-\frac {\sqrt {3}}{12}-\frac {i}{12}\right ) \left (\frac {\left (-\frac {i}{2}-\frac {\sqrt {3}}{2}\right ) {\mathrm e}^{\left (-\frac {i}{2}-\frac {\sqrt {3}}{2}\right ) x} \cos \left (x \right )}{2 \left (-\frac {i}{2}-\frac {\sqrt {3}}{2}\right )^{2}+2}+\frac {{\mathrm e}^{\left (-\frac {i}{2}-\frac {\sqrt {3}}{2}\right ) x} \sin \left (x \right )}{2 \left (-\frac {i}{2}-\frac {\sqrt {3}}{2}\right )^{2}+2}-\frac {\left (-\frac {i}{2}-\frac {\sqrt {3}}{2}\right ) {\mathrm e}^{\left (-\frac {i}{2}-\frac {\sqrt {3}}{2}\right ) x} \cos \left (2 x \right )}{2 \left (\left (-\frac {i}{2}-\frac {\sqrt {3}}{2}\right )^{2}+4\right )}-\frac {{\mathrm e}^{\left (-\frac {i}{2}-\frac {\sqrt {3}}{2}\right ) x} \sin \left (2 x \right )}{\left (-\frac {i}{2}-\frac {\sqrt {3}}{2}\right )^{2}+4}\right )\right ) \left ({\mathrm e}^{\frac {\sqrt {2+2 i \sqrt {3}}\, x}{2}}\right ) \end {split} \end {equation*} Therefore the particular solution is \[ y_p = \frac {{\mathrm e}^{i x}}{12}+\frac {{\mathrm e}^{-i x}}{12}-\frac {i \left (\int \sin \left (\frac {3 x}{2}\right ) \sin \left (\frac {x}{2}\right ) {\mathrm e}^{-i x}d x \right ) {\mathrm e}^{i x}}{6}+\frac {i \left (\int \sin \left (\frac {3 x}{2}\right ) \sin \left (\frac {x}{2}\right ) {\mathrm e}^{i x}d x \right ) {\mathrm e}^{-i x}}{6}-\frac {\cos \left (2 x \right )}{21} \] Which simpliﬁes to \[ y_p = \frac {\cos \left (x \right )}{6}-\frac {\left (\int \sin \left (\frac {3 x}{2}\right ) \sin \left (\frac {x}{2}\right ) \sin \left (x \right )d x \right ) \cos \left (x \right )}{3}+\frac {\left (\int \sin \left (\frac {3 x}{2}\right ) \sin \left (\frac {x}{2}\right ) \cos \left (x \right )d x \right ) \sin \left (x \right )}{3}-\frac {\cos \left (2 x \right )}{21} \] Therefore the general solution is \begin{align*} y &= y_h + y_p \\ &= \left ({\mathrm e}^{-\frac {\sqrt {2+2 i \sqrt {3}}\, x}{2}} c_{1} +{\mathrm e}^{i x} c_{2} +{\mathrm e}^{\frac {\sqrt {2-2 i \sqrt {3}}\, x}{2}} c_{3} +{\mathrm e}^{-i x} c_{4} +{\mathrm e}^{-\frac {\sqrt {2-2 i \sqrt {3}}\, x}{2}} c_{5} +{\mathrm e}^{\frac {\sqrt {2+2 i \sqrt {3}}\, x}{2}} c_{6}\right ) + \left (\frac {\cos \left (x \right )}{6}-\frac {\left (\int \sin \left (\frac {3 x}{2}\right ) \sin \left (\frac {x}{2}\right ) \sin \left (x \right )d x \right ) \cos \left (x \right )}{3}+\frac {\left (\int \sin \left (\frac {3 x}{2}\right ) \sin \left (\frac {x}{2}\right ) \cos \left (x \right )d x \right ) \sin \left (x \right )}{3}-\frac {\cos \left (2 x \right )}{21}\right ) \\ \end{align*} Which simpliﬁes to \[ y = {\mathrm e}^{-\frac {\left (\sqrt {3}+i\right ) x}{2}} c_{1} +{\mathrm e}^{i x} c_{2} +{\mathrm e}^{-\frac {\left (i-\sqrt {3}\right ) x}{2}} c_{3} +{\mathrm e}^{-i x} c_{4} +{\mathrm e}^{\frac {\left (i-\sqrt {3}\right ) x}{2}} c_{5} +{\mathrm e}^{\frac {\left (\sqrt {3}+i\right ) x}{2}} c_{6} +\frac {\cos \left (x \right )}{6}-\frac {\left (\int \sin \left (\frac {3 x}{2}\right ) \sin \left (\frac {x}{2}\right ) \sin \left (x \right )d x \right ) \cos \left (x \right )}{3}+\frac {\left (\int \sin \left (\frac {3 x}{2}\right ) \sin \left (\frac {x}{2}\right ) \cos \left (x \right )d x \right ) \sin \left (x \right )}{3}-\frac {\cos \left (2 x \right )}{21} \]

Summary

The solution(s) found are the following \begin{align*} \tag{1} y &= {\mathrm e}^{-\frac {\left (\sqrt {3}+i\right ) x}{2}} c_{1} +{\mathrm e}^{i x} c_{2} +{\mathrm e}^{-\frac {\left (i-\sqrt {3}\right ) x}{2}} c_{3} +{\mathrm e}^{-i x} c_{4} +{\mathrm e}^{\frac {\left (i-\sqrt {3}\right ) x}{2}} c_{5} +{\mathrm e}^{\frac {\left (\sqrt {3}+i\right ) x}{2}} c_{6} +\frac {\cos \left (x \right )}{6}-\frac {\left (\int \sin \left (\frac {3 x}{2}\right ) \sin \left (\frac {x}{2}\right ) \sin \left (x \right )d x \right ) \cos \left (x \right )}{3}+\frac {\left (\int \sin \left (\frac {3 x}{2}\right ) \sin \left (\frac {x}{2}\right ) \cos \left (x \right )d x \right ) \sin \left (x \right )}{3}-\frac {\cos \left (2 x \right )}{21} \\ \end{align*}

Veriﬁcation of solutions

\[ y = {\mathrm e}^{-\frac {\left (\sqrt {3}+i\right ) x}{2}} c_{1} +{\mathrm e}^{i x} c_{2} +{\mathrm e}^{-\frac {\left (i-\sqrt {3}\right ) x}{2}} c_{3} +{\mathrm e}^{-i x} c_{4} +{\mathrm e}^{\frac {\left (i-\sqrt {3}\right ) x}{2}} c_{5} +{\mathrm e}^{\frac {\left (\sqrt {3}+i\right ) x}{2}} c_{6} +\frac {\cos \left (x \right )}{6}-\frac {\left (\int \sin \left (\frac {3 x}{2}\right ) \sin \left (\frac {x}{2}\right ) \sin \left (x \right )d x \right ) \cos \left (x \right )}{3}+\frac {\left (\int \sin \left (\frac {3 x}{2}\right ) \sin \left (\frac {x}{2}\right ) \cos \left (x \right )d x \right ) \sin \left (x \right )}{3}-\frac {\cos \left (2 x \right )}{21} \] 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] 
trying high order linear exact nonhomogeneous 
trying differential order: 6; missing the dependent variable 
checking if the LODE has constant coefficients 
<- constant coefficients successful`

\[ y \left (x \right ) = \left (\sin \left (\frac {x}{2}\right ) c_{4} +\cos \left (\frac {x}{2}\right ) c_{3} \right ) {\mathrm e}^{-\frac {\sqrt {3}\, x}{2}}+\left (\sin \left (\frac {x}{2}\right ) c_{6} +c_{5} \cos \left (\frac {x}{2}\right )\right ) {\mathrm e}^{\frac {\sqrt {3}\, x}{2}}+\frac {\cos \left (2 x \right )}{126}+\frac {\left (5+24 c_{1} \right ) \cos \left (x \right )}{24}+\frac {\sin \left (x \right ) \left (x +12 c_{2} \right )}{12} \]

\[ y(x)\to \frac {1}{12} x \sin (x)+\frac {1}{126} \cos (2 x)+e^{-\frac {\sqrt {3} x}{2}} \left (c_1 e^{\sqrt {3} x}+c_3\right ) \cos \left (\frac {x}{2}\right )+\left (\frac {1}{4}+c_2\right ) \cos (x)+c_4 e^{-\frac {\sqrt {3} x}{2}} \sin \left (\frac {x}{2}\right )+c_6 e^{\frac {\sqrt {3} x}{2}} \sin \left (\frac {x}{2}\right )+c_5 \sin (x) \]

6.3 problem 1580