2.3.19 problem 19

Internal problem ID [8553]
Book : Own collection of miscellaneous problems
Section : section 3.0
Problem number : 19
Date solved : Thursday, December 12, 2024 at 09:30:02 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

Solve

\begin{align*} x^{4} y^{\prime \prime \prime }+x^{3} y^{\prime \prime }+x^{2} y^{\prime }+x y&=x \end{align*}

Solved as higher order Euler type ode

Time used: 3.369 (sec)

The ode can be normalized and rewritten as Euler ode.

This is Euler ODE of higher order. Let \(y = x^{\lambda }\). Hence

\begin{align*} y^{\prime } &= \lambda \,x^{\lambda -1}\\ y^{\prime \prime } &= \lambda \left (\lambda -1\right ) x^{\lambda -2}\\ y^{\prime \prime \prime } &= \lambda \left (\lambda -1\right ) \left (\lambda -2\right ) x^{\lambda -3} \end{align*}

Substituting these back into

\[ x^{4} y^{\prime \prime \prime }+x^{3} y^{\prime \prime }+x^{2} y^{\prime }+x y = x \]

gives

\[ x \lambda \,x^{\lambda -1}+x^{2} \lambda \left (\lambda -1\right ) x^{\lambda -2}+x^{3} \lambda \left (\lambda -1\right ) \left (\lambda -2\right ) x^{\lambda -3}+x^{\lambda } = 0 \]

Which simplifies to

\[ \lambda \,x^{\lambda }+\lambda \left (\lambda -1\right ) x^{\lambda }+\lambda \left (\lambda -1\right ) \left (\lambda -2\right ) x^{\lambda }+x^{\lambda } = 0 \]

And since \(x^{\lambda }\neq 0\) then dividing through by \(x^{\lambda }\), the above becomes

\[ \lambda +\lambda \left (\lambda -1\right )+\lambda \left (\lambda -1\right ) \left (\lambda -2\right )+1 = 0 \]

Simplifying gives the characteristic equation as

\[ \lambda ^{3}-2 \lambda ^{2}+2 \lambda +1 = 0 \]

Solving the above gives the following roots

\begin{align*} \lambda _1 &= -\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}+\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}\\ \lambda _2 &= \frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}+\frac {i \sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right )}{2}\\ \lambda _3 &= \frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}-\frac {i \sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right )}{2} \end{align*}

This table summarises the result

root	multiplicity	type of root
\(-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}+\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}\)	\(1\)	real root
\(\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3} \pm -\frac {\sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right )}{2} i\)	\(1\)	complex conjugate root

The solution is generated by going over the above table. For each real root \(\lambda \) of multiplicity one generates a \(c_1x^{\lambda }\) basis solution. Each real root of multiplicty two, generates \(c_1x^{\lambda }\) and \(c_2x^{\lambda } \ln \left (x \right )\) basis solutions. Each real root of multiplicty three, generates \(c_1x^{\lambda }\) and \(c_2x^{\lambda } \ln \left (x \right )\) and \(c_3x^{\lambda } \ln \left (x \right )^{2}\) basis solutions, and so on. Each complex root \(\alpha \pm i \beta \) of multiplicity one generates \(x^{\alpha } \left (c_1\cos (\beta \ln \left (x \right ))+c_2\sin (\beta \ln \left (x \right ))\right )\) basis solutions. And each complex root \(\alpha \pm i \beta \) of multiplicity two generates \(\ln \left (x \right ) x^{\alpha }\left (c_1\cos (\beta \ln \left (x \right ))+c_2\sin (\beta \ln \left (x \right ))\right )\) basis solutions. And each complex root \(\alpha \pm i \beta \) of multiplicity three generates \(\ln \left (x \right )^{2} x^{\alpha }\left (c_1\cos (\beta \ln \left (x \right ))+c_2\sin (\beta \ln \left (x \right ))\right )\) basis solutions. And so on. Using the above show that the solution is

\[ y = c_1 \,x^{-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}+\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}}+x^{\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}} \left (c_2 \cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) \ln \left (x \right )}{2}\right )-c_3 \sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) \ln \left (x \right )}{2}\right )\right ) \]

The fundamental set of solutions for the homogeneous solution are the following

\begin{align*} y_1 &= x^{-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}+\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}} \\ y_2 &= x^{\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}} \cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) \ln \left (x \right )}{2}\right ) \\ y_3 &= -x^{\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}} \sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) \ln \left (x \right )}{2}\right ) \\ \end{align*}

This is higher order nonhomogeneous Euler type ODE. Let the solution be

\[ y = y_h + y_p \]

Where \(y_h\) is the solution to the homogeneous Euler ODE And \(y_p\) is a particular solution to the nonhomogeneous Euler ODE. \(y_h\) is the solution to

\[ y^{\prime } x +y^{\prime \prime } x^{2}+y^{\prime \prime \prime } x^{3}+y = 0 \]

Now the particular solution to the given ODE is found

\[ y^{\prime } x +y^{\prime \prime } x^{2}+y^{\prime \prime \prime } x^{3}+y = x \]

Let the particular solution be

\[ y_p = U_1 y_1+U_2 y_2+U_3 y_3 \]

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_1'&y_2'&y_3'\\ y_1''&y_2''&y_3''\\ \end {vmatrix} \end{equation*}

Substituting the fundamental set of solutions \(y_i\) found above in the Wronskian gives

\begin{align*} W &= \text {Expression too large to display} \\ |W| &= \text {Expression too large to display} \end{align*}

The determinant simplifies to

\begin{align*} |W| &= \frac {\sqrt {83}}{2 x} \end{align*}

Now we determine \(W_i\) for each \(U_i\).

\begin{align*} W_1(x) &= \det \,\left [\begin {array}{cc} \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+8\right ) \ln \left (x \right )}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}\right ) x^{\frac {\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8 \left (188+12 \sqrt {249}\right )^{{1}/{3}}-8}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}} & x^{\frac {\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8 \left (188+12 \sqrt {249}\right )^{{1}/{3}}-8}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}} \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+8\right ) \ln \left (x \right )}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}\right ) \\ -\frac {x^{\frac {\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}-4 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}-8}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}} \left (\left (-\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}-8 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+8\right ) \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+8\right ) \ln \left (x \right )}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}\right )+\sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+8\right ) \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+8\right ) \ln \left (x \right )}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}\right )\right )}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}} & \frac {x^{\frac {\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}-4 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}-8}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}} \left (\left (\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+8 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}-8\right ) \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+8\right ) \ln \left (x \right )}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}\right )+\sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+8\right ) \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+8\right ) \ln \left (x \right )}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}\right )\right )}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}} \end {array}\right ] \\ &= \frac {\sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+8\right ) x^{\frac {\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}{6}+\frac {1}{3}-\frac {4}{3 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}}}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}} \end{align*}

\begin{align*} W_2(x) &= \det \,\left [\begin {array}{cc} x^{-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}+\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}} & x^{\frac {\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8 \left (188+12 \sqrt {249}\right )^{{1}/{3}}-8}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}} \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+8\right ) \ln \left (x \right )}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}\right ) \\ \frac {x^{-\frac {\left (\left (188+12 \sqrt {249}\right )^{{1}/{3}}+4\right ) \left (\left (188+12 \sqrt {249}\right )^{{1}/{3}}-2\right )}{6 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}} \left (2-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{2}+\frac {4}{\left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right )}{3} & \frac {x^{\frac {\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}-4 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}-8}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}} \left (\left (\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+8 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}-8\right ) \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+8\right ) \ln \left (x \right )}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}\right )+\sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+8\right ) \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+8\right ) \ln \left (x \right )}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}\right )\right )}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}} \end {array}\right ] \\ &= -\frac {3 \left (\left (\left (\frac {2 i}{3}-\frac {2 \sqrt {3}}{9}\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}-i \sqrt {3}\, \sqrt {83}-\frac {47 i}{3}-\frac {47 \sqrt {3}}{9}-\sqrt {83}\right ) x^{\frac {-2 i \sqrt {3}\, \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}-3 \sqrt {3}\, \sqrt {83}-47 i \sqrt {3}-9 i \sqrt {83}+2 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}-47}{3 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}}+\frac {x^{\frac {2 i \sqrt {3}\, \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+9 i \sqrt {83}+47 i \sqrt {3}-3 \sqrt {3}\, \sqrt {83}+\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+2 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}-47}{3 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}} \left (\left (-6 i-2 \sqrt {3}\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+9 i \sqrt {3}\, \sqrt {83}+141 i-47 \sqrt {3}-9 \sqrt {83}\right )}{9}\right )}{2 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}} \end{align*}

\begin{align*} W_3(x) &= \det \,\left [\begin {array}{cc} x^{-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}+\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}} & \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+8\right ) \ln \left (x \right )}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}\right ) x^{\frac {\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8 \left (188+12 \sqrt {249}\right )^{{1}/{3}}-8}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}} \\ \frac {x^{-\frac {\left (\left (188+12 \sqrt {249}\right )^{{1}/{3}}+4\right ) \left (\left (188+12 \sqrt {249}\right )^{{1}/{3}}-2\right )}{6 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}} \left (2-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{2}+\frac {4}{\left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right )}{3} & -\frac {x^{\frac {\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}-4 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}-8}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}} \left (\left (-\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}-8 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+8\right ) \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+8\right ) \ln \left (x \right )}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}\right )+\sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+8\right ) \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+8\right ) \ln \left (x \right )}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}\right )\right )}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}} \end {array}\right ] \\ &= \frac {\left (\frac {141}{2}+\left (-i \sqrt {3}-3\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+\frac {\left (-47 i+9 \sqrt {83}\right ) \sqrt {3}}{2}-\frac {9 i \sqrt {83}}{2}\right ) x^{\frac {-2 i \sqrt {3}\, \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}-3 \sqrt {3}\, \sqrt {83}-47 i \sqrt {3}-9 i \sqrt {83}+2 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}-47}{3 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}}+\frac {x^{\frac {2 i \sqrt {3}\, \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+9 i \sqrt {83}+47 i \sqrt {3}-3 \sqrt {3}\, \sqrt {83}+\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+2 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}-47}{3 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}} \left (141+2 \left (-3+i \sqrt {3}\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+\sqrt {3}\, \left (47 i+9 \sqrt {83}\right )+9 i \sqrt {83}\right )}{2}}{3 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}} \end{align*}

Now we are ready to evaluate each \(U_i(x)\).

\begin{align*} U_1 &= (-1)^{3-1} \int { \frac {F(x) W_1(x) }{a W(x)} \, dx}\\ &= (-1)^{2} \int { \frac { \left (x\right ) \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+8\right ) x^{\frac {\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}{6}+\frac {1}{3}-\frac {4}{3 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}}}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}\right )}{\left (x^{3}\right ) \left (\frac {\sqrt {83}}{2 x}\right )} \, dx} \\ &= \int { \frac {\frac {x \sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+8\right ) x^{\frac {\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}{6}+\frac {1}{3}-\frac {4}{3 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}}}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}}{\frac {x^{2} \sqrt {83}}{2}} \, dx}\\ &= \int {\left (\frac {x^{\frac {\left (188+12 \sqrt {249}\right )^{{2}/{3}}-4 \left (188+12 \sqrt {249}\right )^{{1}/{3}}-8}{6 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}} \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) \sqrt {249}}{498 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) \, dx} \\ &= \frac {x^{1+\frac {\left (188+12 \sqrt {249}\right )^{{2}/{3}}-4 \left (188+12 \sqrt {249}\right )^{{1}/{3}}-8}{6 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}} \sqrt {249}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) \left (\sqrt {249}\, \left (188+12 \sqrt {249}\right )^{{2}/{3}}-13 \left (188+12 \sqrt {249}\right )^{{2}/{3}}+4 \left (188+12 \sqrt {249}\right )^{{1}/{3}} \sqrt {249}-68 \left (188+12 \sqrt {249}\right )^{{1}/{3}}+32\right )}{95616 \left (188+12 \sqrt {249}\right )^{{1}/{3}}} \\ &= \frac {x^{1+\frac {\left (188+12 \sqrt {249}\right )^{{2}/{3}}-4 \left (188+12 \sqrt {249}\right )^{{1}/{3}}-8}{6 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}} \sqrt {249}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) \left (\sqrt {249}\, \left (188+12 \sqrt {249}\right )^{{2}/{3}}-13 \left (188+12 \sqrt {249}\right )^{{2}/{3}}+4 \left (188+12 \sqrt {249}\right )^{{1}/{3}} \sqrt {249}-68 \left (188+12 \sqrt {249}\right )^{{1}/{3}}+32\right )}{95616 \left (188+12 \sqrt {249}\right )^{{1}/{3}}} \end{align*}

\begin{align*} U_2 &= (-1)^{3-2} \int { \frac {F(x) W_2(x) }{a W(x)} \, dx}\\ &= (-1)^{1} \int { \frac { \left (x\right ) \left (-\frac {3 \left (\left (\left (\frac {2 i}{3}-\frac {2 \sqrt {3}}{9}\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}-i \sqrt {3}\, \sqrt {83}-\frac {47 i}{3}-\frac {47 \sqrt {3}}{9}-\sqrt {83}\right ) x^{\frac {-2 i \sqrt {3}\, \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}-3 \sqrt {3}\, \sqrt {83}-47 i \sqrt {3}-9 i \sqrt {83}+2 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}-47}{3 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}}+\frac {x^{\frac {2 i \sqrt {3}\, \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+9 i \sqrt {83}+47 i \sqrt {3}-3 \sqrt {3}\, \sqrt {83}+\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+2 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}-47}{3 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}} \left (\left (-6 i-2 \sqrt {3}\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+9 i \sqrt {3}\, \sqrt {83}+141 i-47 \sqrt {3}-9 \sqrt {83}\right )}{9}\right )}{2 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}\right )}{\left (x^{3}\right ) \left (\frac {\sqrt {83}}{2 x}\right )} \, dx} \\ &= - \int { \frac {-\frac {3 x \left (\left (\left (\frac {2 i}{3}-\frac {2 \sqrt {3}}{9}\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}-i \sqrt {3}\, \sqrt {83}-\frac {47 i}{3}-\frac {47 \sqrt {3}}{9}-\sqrt {83}\right ) x^{\frac {-2 i \sqrt {3}\, \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}-3 \sqrt {3}\, \sqrt {83}-47 i \sqrt {3}-9 i \sqrt {83}+2 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}-47}{3 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}}+\frac {x^{\frac {2 i \sqrt {3}\, \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+9 i \sqrt {83}+47 i \sqrt {3}-3 \sqrt {3}\, \sqrt {83}+\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+2 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}-47}{3 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}} \left (\left (-6 i-2 \sqrt {3}\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+9 i \sqrt {3}\, \sqrt {83}+141 i-47 \sqrt {3}-9 \sqrt {83}\right )}{9}\right )}{2 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}}{\frac {x^{2} \sqrt {83}}{2}} \, dx}\\ &= - \int {\left (-\frac {3 \left (\left (\left (\frac {2 i}{3}-\frac {2 \sqrt {3}}{9}\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}-i \sqrt {3}\, \sqrt {83}-\frac {47 i}{3}-\frac {47 \sqrt {3}}{9}-\sqrt {83}\right ) x^{\frac {-2 i \sqrt {3}\, \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}-3 \sqrt {3}\, \sqrt {83}-47 i \sqrt {3}-9 i \sqrt {83}+2 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}-47}{3 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}}+\frac {x^{\frac {2 i \sqrt {3}\, \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+9 i \sqrt {83}+47 i \sqrt {3}-3 \sqrt {3}\, \sqrt {83}+\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+2 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}-47}{3 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}} \left (\left (-6 i-2 \sqrt {3}\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+9 i \sqrt {3}\, \sqrt {83}+141 i-47 \sqrt {3}-9 \sqrt {83}\right )}{9}\right ) \sqrt {83}}{83 x \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}\right ) \, dx}\\ &= -\frac {\left (-\frac {3 i \sqrt {83}}{332}-\frac {\sqrt {3}\, \sqrt {83}}{332}-\frac {i \sqrt {3}}{12}-\frac {1}{12}-\frac {\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}{6}-\frac {i \sqrt {3}\, \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}{96}+\frac {3 i \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}} \sqrt {83}}{2656}+\frac {\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}{96}-\frac {\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}} \sqrt {3}\, \sqrt {83}}{2656}\right ) {\mathrm e}^{\frac {\left (2 i \sqrt {3}\, \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+9 i \sqrt {83}+47 i \sqrt {3}-3 \sqrt {3}\, \sqrt {83}+\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+2 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}-47\right ) \ln \left (x \right )}{3 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}}+\left (\frac {3 i \sqrt {83}}{332}-\frac {\sqrt {3}\, \sqrt {83}}{332}+\frac {i \sqrt {3}}{12}-\frac {1}{12}-\frac {\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}{6}+\frac {i \sqrt {3}\, \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}{96}-\frac {3 i \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}} \sqrt {83}}{2656}+\frac {\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}{96}-\frac {\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}} \sqrt {3}\, \sqrt {83}}{2656}\right ) {\mathrm e}^{\frac {\left (-2 i \sqrt {3}\, \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}-3 \sqrt {3}\, \sqrt {83}-47 i \sqrt {3}-9 i \sqrt {83}+2 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}-47\right ) \ln \left (x \right )}{3 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}}}{\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}} \end{align*}

\begin{align*} U_3 &= (-1)^{3-3} \int { \frac {F(x) W_3(x) }{a W(x)} \, dx}\\ &= (-1)^{0} \int { \frac { \left (x\right ) \left (\frac {\left (\frac {141}{2}+\left (-i \sqrt {3}-3\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+\frac {\left (-47 i+9 \sqrt {83}\right ) \sqrt {3}}{2}-\frac {9 i \sqrt {83}}{2}\right ) x^{\frac {-2 i \sqrt {3}\, \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}-3 \sqrt {3}\, \sqrt {83}-47 i \sqrt {3}-9 i \sqrt {83}+2 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}-47}{3 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}}+\frac {x^{\frac {2 i \sqrt {3}\, \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+9 i \sqrt {83}+47 i \sqrt {3}-3 \sqrt {3}\, \sqrt {83}+\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+2 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}-47}{3 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}} \left (141+2 \left (-3+i \sqrt {3}\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+\sqrt {3}\, \left (47 i+9 \sqrt {83}\right )+9 i \sqrt {83}\right )}{2}}{3 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}\right )}{\left (x^{3}\right ) \left (\frac {\sqrt {83}}{2 x}\right )} \, dx} \\ &= \int { \frac {\frac {x \left (\left (\frac {141}{2}+\left (-i \sqrt {3}-3\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+\frac {\left (-47 i+9 \sqrt {83}\right ) \sqrt {3}}{2}-\frac {9 i \sqrt {83}}{2}\right ) x^{\frac {-2 i \sqrt {3}\, \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}-3 \sqrt {3}\, \sqrt {83}-47 i \sqrt {3}-9 i \sqrt {83}+2 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}-47}{3 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}}+\frac {x^{\frac {2 i \sqrt {3}\, \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+9 i \sqrt {83}+47 i \sqrt {3}-3 \sqrt {3}\, \sqrt {83}+\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+2 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}-47}{3 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}} \left (141+2 \left (-3+i \sqrt {3}\right ) \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+\sqrt {3}\, \left (47 i+9 \sqrt {83}\right )+9 i \sqrt {83}\right )}{2}\right )}{3 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}}{\frac {x^{2} \sqrt {83}}{2}} \, dx}\\ &= \int {\left (-\frac {2 \sqrt {83}\, \left (\left (\left (3+i \sqrt {3}\right ) \left (188+12 \sqrt {249}\right )^{{1}/{3}}+\frac {47 i \sqrt {3}}{2}+\frac {9 i \sqrt {83}}{2}-\frac {9 \sqrt {249}}{2}-\frac {141}{2}\right ) x^{\frac {-2 i \sqrt {3}\, \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}-3 \sqrt {3}\, \sqrt {83}-47 i \sqrt {3}-9 i \sqrt {83}+2 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}-47}{3 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}}-x^{\frac {2 i \sqrt {3}\, \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+9 i \sqrt {83}+47 i \sqrt {3}-3 \sqrt {3}\, \sqrt {83}+\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+2 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}-47}{3 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}} \left (\left (-3+i \sqrt {3}\right ) \left (188+12 \sqrt {249}\right )^{{1}/{3}}+\frac {47 i \sqrt {3}}{2}+\frac {9 i \sqrt {83}}{2}+\frac {9 \sqrt {249}}{2}+\frac {141}{2}\right )\right )}{249 \left (188+12 \sqrt {249}\right )^{{2}/{3}} x}\right ) \, dx} \\ &= \frac {\left (\frac {3 \sqrt {83}}{332}-\frac {i \sqrt {3}\, \sqrt {83}}{332}+\frac {\sqrt {3}}{12}-\frac {i}{12}-\frac {i \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}{6}+\frac {i \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}{96}+\frac {\sqrt {3}\, \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}{96}-\frac {3 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}} \sqrt {83}}{2656}-\frac {i \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}} \sqrt {3}\, \sqrt {83}}{2656}\right ) {\mathrm e}^{\frac {\left (2 i \sqrt {3}\, \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+9 i \sqrt {83}+47 i \sqrt {3}-3 \sqrt {3}\, \sqrt {83}+\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+2 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}-47\right ) \ln \left (x \right )}{3 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}}+\left (\frac {3 \sqrt {83}}{332}+\frac {i \sqrt {3}\, \sqrt {83}}{332}+\frac {\sqrt {3}}{12}+\frac {i}{12}+\frac {i \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}{6}-\frac {i \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}{96}+\frac {\sqrt {3}\, \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}{96}-\frac {3 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}} \sqrt {83}}{2656}+\frac {i \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}} \sqrt {3}\, \sqrt {83}}{2656}\right ) {\mathrm e}^{\frac {\left (-2 i \sqrt {3}\, \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}-3 \sqrt {3}\, \sqrt {83}-47 i \sqrt {3}-9 i \sqrt {83}+2 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}-47\right ) \ln \left (x \right )}{3 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}}}{\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}} \\ &= \frac {\left (\frac {3 \sqrt {83}}{332}-\frac {i \sqrt {3}\, \sqrt {83}}{332}+\frac {\sqrt {3}}{12}-\frac {i}{12}-\frac {i \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}{6}+\frac {i \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}{96}+\frac {\sqrt {3}\, \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}{96}-\frac {3 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}} \sqrt {83}}{2656}-\frac {i \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}} \sqrt {3}\, \sqrt {83}}{2656}\right ) {\mathrm e}^{\frac {\left (2 i \sqrt {3}\, \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+9 i \sqrt {83}+47 i \sqrt {3}-3 \sqrt {3}\, \sqrt {83}+\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+2 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}-47\right ) \ln \left (x \right )}{3 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}}+\left (\frac {3 \sqrt {83}}{332}+\frac {i \sqrt {3}\, \sqrt {83}}{332}+\frac {\sqrt {3}}{12}+\frac {i}{12}+\frac {i \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}{6}-\frac {i \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}{96}+\frac {\sqrt {3}\, \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}{96}-\frac {3 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}} \sqrt {83}}{2656}+\frac {i \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}} \sqrt {3}\, \sqrt {83}}{2656}\right ) {\mathrm e}^{\frac {\left (-2 i \sqrt {3}\, \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}-3 \sqrt {3}\, \sqrt {83}-47 i \sqrt {3}-9 i \sqrt {83}+2 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}-47\right ) \ln \left (x \right )}{3 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}}}{\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}} \end{align*}

Now that all the \(U_i\) functions have been determined, the particular solution is found from

\[ y_p = U_1 y_1+U_2 y_2+U_3 y_3 \]

Hence

\begin{equation*} \begin {split} y_p &= \left (\frac {x^{1+\frac {\left (188+12 \sqrt {249}\right )^{{2}/{3}}-4 \left (188+12 \sqrt {249}\right )^{{1}/{3}}-8}{6 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}} \sqrt {249}\, \left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8\right ) \left (\sqrt {249}\, \left (188+12 \sqrt {249}\right )^{{2}/{3}}-13 \left (188+12 \sqrt {249}\right )^{{2}/{3}}+4 \left (188+12 \sqrt {249}\right )^{{1}/{3}} \sqrt {249}-68 \left (188+12 \sqrt {249}\right )^{{1}/{3}}+32\right )}{95616 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) \left (x^{-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}+\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}}\right ) \\ &+\left (-\frac {\left (-\frac {3 i \sqrt {83}}{332}-\frac {\sqrt {3}\, \sqrt {83}}{332}-\frac {i \sqrt {3}}{12}-\frac {1}{12}-\frac {\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}{6}-\frac {i \sqrt {3}\, \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}{96}+\frac {3 i \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}} \sqrt {83}}{2656}+\frac {\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}{96}-\frac {\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}} \sqrt {3}\, \sqrt {83}}{2656}\right ) {\mathrm e}^{\frac {\left (2 i \sqrt {3}\, \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+9 i \sqrt {83}+47 i \sqrt {3}-3 \sqrt {3}\, \sqrt {83}+\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+2 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}-47\right ) \ln \left (x \right )}{3 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}}+\left (\frac {3 i \sqrt {83}}{332}-\frac {\sqrt {3}\, \sqrt {83}}{332}+\frac {i \sqrt {3}}{12}-\frac {1}{12}-\frac {\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}{6}+\frac {i \sqrt {3}\, \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}{96}-\frac {3 i \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}} \sqrt {83}}{2656}+\frac {\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}{96}-\frac {\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}} \sqrt {3}\, \sqrt {83}}{2656}\right ) {\mathrm e}^{\frac {\left (-2 i \sqrt {3}\, \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}-3 \sqrt {3}\, \sqrt {83}-47 i \sqrt {3}-9 i \sqrt {83}+2 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}-47\right ) \ln \left (x \right )}{3 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}}}{\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}\right ) \left (x^{\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}} \cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) \ln \left (x \right )}{2}\right )\right ) \\ &+\left (\frac {\left (\frac {3 \sqrt {83}}{332}-\frac {i \sqrt {3}\, \sqrt {83}}{332}+\frac {\sqrt {3}}{12}-\frac {i}{12}-\frac {i \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}{6}+\frac {i \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}{96}+\frac {\sqrt {3}\, \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}{96}-\frac {3 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}} \sqrt {83}}{2656}-\frac {i \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}} \sqrt {3}\, \sqrt {83}}{2656}\right ) {\mathrm e}^{\frac {\left (2 i \sqrt {3}\, \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+9 i \sqrt {83}+47 i \sqrt {3}-3 \sqrt {3}\, \sqrt {83}+\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+2 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}-47\right ) \ln \left (x \right )}{3 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}}+\left (\frac {3 \sqrt {83}}{332}+\frac {i \sqrt {3}\, \sqrt {83}}{332}+\frac {\sqrt {3}}{12}+\frac {i}{12}+\frac {i \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}{6}-\frac {i \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}{96}+\frac {\sqrt {3}\, \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}{96}-\frac {3 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}} \sqrt {83}}{2656}+\frac {i \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}} \sqrt {3}\, \sqrt {83}}{2656}\right ) {\mathrm e}^{\frac {\left (-2 i \sqrt {3}\, \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}-3 \sqrt {3}\, \sqrt {83}-47 i \sqrt {3}-9 i \sqrt {83}+2 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}-47\right ) \ln \left (x \right )}{3 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}}}}{\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}\right ) \left (-x^{\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}} \sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) \ln \left (x \right )}{2}\right )\right ) \end {split} \end{equation*}

Therefore the particular solution is

\[ y_p = \frac {x}{2} \]

Therefore the general solution is

\begin{align*} y &= y_h + y_p \\ &= \left (c_1 \,x^{-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}+\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}}+x^{\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}} \left (c_2 \cos \left (\frac {\sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) \ln \left (x \right )}{2}\right )-c_3 \sin \left (\frac {\sqrt {3}\, \left (-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}-\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}\right ) \ln \left (x \right )}{2}\right )\right )\right ) + \left (\frac {x}{2}\right ) \\ \end{align*}

Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{4} \left (\frac {d^{3}}{d x^{3}}y \left (x \right )\right )+\left (\frac {d^{2}}{d x^{2}}y \left (x \right )\right ) x^{3}+x^{2} \left (\frac {d}{d x}y \left (x \right )\right )+x y \left (x \right )=x \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 3 \\ {} & {} & \frac {d^{3}}{d x^{3}}y \left (x \right ) \end {array} \]

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 is of Euler type 
<- LODE of Euler type successful 
Euler equation successful`
 

Maple dsolve solution

Solving time : 0.013 (sec)
Leaf size : 223

dsolve(x^4*diff(diff(diff(y(x),x),x),x)+x^3*diff(diff(y(x),x),x)+diff(y(x),x)*x^2+x*y(x) = x, 
       y(x),singsol=all)
 

\[ y = c_{2} x^{\frac {\left (47-3 \sqrt {249}\right ) \left (188+12 \sqrt {249}\right )^{{2}/{3}}}{192}+\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{12}+\frac {2}{3}} \cos \left (\frac {\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}} \sqrt {3}\, \left (3 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}} \sqrt {3}\, \sqrt {83}-47 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+16\right ) \ln \left (x \right )}{192}\right )+c_3 \,x^{\frac {\left (47-3 \sqrt {249}\right ) \left (188+12 \sqrt {249}\right )^{{2}/{3}}}{192}+\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{12}+\frac {2}{3}} \sin \left (\frac {\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}} \sqrt {3}\, \left (3 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}} \sqrt {3}\, \sqrt {83}-47 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+16\right ) \ln \left (x \right )}{192}\right )+x^{\frac {\left (-47+3 \sqrt {249}\right ) \left (188+12 \sqrt {249}\right )^{{2}/{3}}}{96}-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}+\frac {2}{3}} c_{1} +1 \]

Mathematica DSolve solution

Solving time : 0.004 (sec)
Leaf size : 82

DSolve[{x^4*D[y[x],{x,3}]+x^3*D[y[x],{x,2}]+x^2*D[y[x],x]+x*y[x]== x,{}}, 
       y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_1 x^{\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+2 \text {$\#$1}+1\&,1\right ]}+c_3 x^{\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+2 \text {$\#$1}+1\&,3\right ]}+c_2 x^{\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+2 \text {$\#$1}+1\&,2\right ]}+1 \]