2.3.19 Problem 19

Internal problem ID [8876]
Book : Own collection of miscellaneous problems
Section : section 3.0
Problem number : 19
Date solved : Friday, April 25, 2025 at 05:16:30 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

Solved as higher order Euler type ode

Time used: 4.576 (sec)

The ode can be normalized and rewritten as Euler ode.

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

$$\begin{array}{rl}{y}^{\prime }& =\lambda x^{\lambda -1}\\ y^{\prime \prime }& =\lambda (\lambda -1)x^{\lambda -2}\\ y^{\prime \prime \prime }& =\lambda (\lambda -1)(\lambda -2)x^{\lambda -3}\end{array}$$

Substituting these back into

$$x^4{y}^{\prime \prime \prime }+x^3{y}^{\prime \prime }+x^2{y}^{\prime }+xy=x$$

gives

$$x\lambda x^{\lambda -1}+x^2\lambda (\lambda -1)x^{\lambda -2}+x^3\lambda (\lambda -1)(\lambda -2)x^{\lambda -3}+x^{\lambda }=0$$

Which simplifies to

$$\lambda x^{\lambda }+\lambda (\lambda -1)x^{\lambda }+\lambda (\lambda -1)(\lambda -2)x^{\lambda }+x^{\lambda }=0$$

And since $x^{\lambda }\ne 0$ then dividing through by $x^{\lambda }$, the above becomes

$$\lambda +\lambda (\lambda -1)+\lambda (\lambda -1)(\lambda -2)+1=0$$

Simplifying gives the characteristic equation as

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

Solving the above gives the following roots

$$\begin{array}{rl}{\lambda }_1& =-\frac{{(188+12\sqrt{249})}^{1/3}}{6}+\frac{4}{3{(188+12\sqrt{249})}^{1/3}}+\frac{2}{3}\\ {\lambda }_2& =\frac{{(188+12\sqrt{249})}^{1/3}}{12}-\frac{2}{3{(188+12\sqrt{249})}^{1/3}}+\frac{2}{3}+\frac{i\sqrt{3}(-\frac{{(188+12\sqrt{249})}^{1/3}}{6}-\frac{4}{3{(188+12\sqrt{249})}^{1/3}})}{2}\\ {\lambda }_3& =\frac{{(188+12\sqrt{249})}^{1/3}}{12}-\frac{2}{3{(188+12\sqrt{249})}^{1/3}}+\frac{2}{3}-\frac{i\sqrt{3}(-\frac{{(188+12\sqrt{249})}^{1/3}}{6}-\frac{4}{3{(188+12\sqrt{249})}^{1/3}})}{2}\end{array}$$

This table summarises the result

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

The solution is generated by going over the above table. For each real root $\lambda$ of multiplicity one generates a $c_1{x}^{\lambda }$ basis solution. Each real root of multiplicty two, generates $c_1{x}^{\lambda }$ and $c_2{x}^{\lambda }\ln \left(x\right)$ basis solutions. Each real root of multiplicty three, generates $c_1{x}^{\lambda }$ and $c_2{x}^{\lambda }\ln \left(x\right)$ and $c_3{x}^{\lambda }\ln {\left(x\right)}^2$ basis solutions, and so on. Each complex root $\alpha \pm i\beta$ of multiplicity one generates $x^{\alpha }(c_1\cos (\beta \ln \left(x\right))+c_2\sin (\beta \ln \left(x\right)))$ basis solutions. And each complex root $\alpha \pm i\beta$ of multiplicity two generates $\ln \left(x\right)x^{\alpha }(c_1\cos (\beta \ln \left(x\right))+c_2\sin (\beta \ln \left(x\right)))$ basis solutions. And each complex root $\alpha \pm i\beta$ of multiplicity three generates $\ln {\left(x\right)}^2{x}^{\alpha }(c_1\cos (\beta \ln \left(x\right))+c_2\sin (\beta \ln \left(x\right)))$ basis solutions. And so on. Using the above show that the solution is

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

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

$$\begin{array}{rl}{y}_1& =x^{\frac{{(188+12\sqrt{249})}^{1/3}}{12}-\frac{2}{3{(188+12\sqrt{249})}^{1/3}}+\frac{2}{3}}\cos \left(\frac{\sqrt{3}(-\frac{{(188+12\sqrt{249})}^{1/3}}{6}-\frac{4}{3{(188+12\sqrt{249})}^{1/3}})\ln \left(x\right)}{2}\right)\\ y_2& =-x^{\frac{{(188+12\sqrt{249})}^{1/3}}{12}-\frac{2}{3{(188+12\sqrt{249})}^{1/3}}+\frac{2}{3}}\sin \left(\frac{\sqrt{3}(-\frac{{(188+12\sqrt{249})}^{1/3}}{6}-\frac{4}{3{(188+12\sqrt{249})}^{1/3}})\ln \left(x\right)}{2}\right)\\ y_3& =x^{-\frac{{(188+12\sqrt{249})}^{1/3}}{6}+\frac{4}{3{(188+12\sqrt{249})}^{1/3}}+\frac{2}{3}}\end{array}$$

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

$$U_i=(-1{)}^{n-i}\int \frac{F(x)W_i(x)}{aW(x)}dx$$

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

$$W(x)=\left|\begin{array}{ccc}{y}_1& y_2& y_3\\ y_1^{\prime }& y_2^{\prime }& y_3^{\prime }\\ y_1^{″}& y_2^{″}& y_3^{″}\end{array}\right|$$

Substituting the fundamental set of solutions $y_i$ found above in the Wronskian gives

$$\begin{array}{rl}W& =\text{Expression too large to display}\\ |W|& =\text{Expression too large to display}\end{array}$$

The determinant simplifies to

$$\begin{array}{rl}|W|& =\frac{\sqrt{83}}{2x}\end{array}$$

Now we determine $W_i$ for each $U_i$.

$$\begin{array}{rl}{W}_1(x)& =det\left[\begin{array}{cc}{x}^{\frac{{(188+12\sqrt{249})}^{2/3}+8{(188+12\sqrt{249})}^{1/3}-8}{12{(188+12\sqrt{249})}^{1/3}}}\sin \left(\frac{\sqrt{3}({(188+12\sqrt{3}\sqrt{83})}^{2/3}+8)\ln \left(x\right)}{12{(188+12\sqrt{3}\sqrt{83})}^{1/3}}\right)& x^{-\frac{{(188+12\sqrt{249})}^{1/3}}{6}+\frac{4}{3{(188+12\sqrt{249})}^{1/3}}+\frac{2}{3}}\\ \frac{x^{\frac{{(188+12\sqrt{3}\sqrt{83})}^{2/3}-4{(188+12\sqrt{3}\sqrt{83})}^{1/3}-8}{12{(188+12\sqrt{3}\sqrt{83})}^{1/3}}}(({(188+12\sqrt{3}\sqrt{83})}^{2/3}+8{(188+12\sqrt{3}\sqrt{83})}^{1/3}-8)\sin \left(\frac{\sqrt{3}({(188+12\sqrt{3}\sqrt{83})}^{2/3}+8)\ln \left(x\right)}{12{(188+12\sqrt{3}\sqrt{83})}^{1/3}}\right)+\sqrt{3}({(188+12\sqrt{3}\sqrt{83})}^{2/3}+8)\cos \left(\frac{\sqrt{3}({(188+12\sqrt{3}\sqrt{83})}^{2/3}+8)\ln \left(x\right)}{12{(188+12\sqrt{3}\sqrt{83})}^{1/3}}\right))}{12{(188+12\sqrt{3}\sqrt{83})}^{1/3}}& -\frac{x^{-\frac{{(188+12\sqrt{249})}^{2/3}+2{(188+12\sqrt{249})}^{1/3}-8}{6{(188+12\sqrt{249})}^{1/3}}}({(188+12\sqrt{249})}^{2/3}-4{(188+12\sqrt{249})}^{1/3}-8)}{6{(188+12\sqrt{249})}^{1/3}}\end{array}\right]\\ & =\frac{\frac{3((-\frac{2i}{3}-\frac{2\sqrt{3}}{9}){(188+12\sqrt{249})}^{1/3}+i\sqrt{249}+\frac{47i}{3}-\frac{47\sqrt{3}}{9}-\sqrt{83})x^{\frac{2i\sqrt{3}{(188+12\sqrt{3}\sqrt{83})}^{1/3}+47i\sqrt{3}+9i\sqrt{83}-3\sqrt{3}\sqrt{83}+{(188+12\sqrt{3}\sqrt{83})}^{2/3}+2{(188+12\sqrt{3}\sqrt{83})}^{1/3}-47}{3{(188+12\sqrt{3}\sqrt{83})}^{2/3}}}}{2}-\frac{3x^{\frac{(-2i\sqrt{3}+2){(188+12\sqrt{3}\sqrt{83})}^{1/3}+{(188+12\sqrt{3}\sqrt{83})}^{2/3}+(-9i-3\sqrt{3})\sqrt{83}-47i\sqrt{3}-47}{3{(188+12\sqrt{3}\sqrt{83})}^{2/3}}}((-\frac{2i}{3}+\frac{2\sqrt{3}}{9}){(188+12\sqrt{249})}^{1/3}+i\sqrt{249}+\frac{47i}{3}+\frac{47\sqrt{3}}{9}+\sqrt{83})}{2}}{{(188+12\sqrt{249})}^{2/3}}\end{array}$$

$$\begin{array}{rl}{W}_2(x)& =det\left[\begin{array}{cc}{x}^{\frac{{(188+12\sqrt{249})}^{2/3}+8{(188+12\sqrt{249})}^{1/3}-8}{12{(188+12\sqrt{249})}^{1/3}}}\cos \left(\frac{\sqrt{3}({(188+12\sqrt{3}\sqrt{83})}^{2/3}+8)\ln \left(x\right)}{12{(188+12\sqrt{3}\sqrt{83})}^{1/3}}\right)& x^{-\frac{{(188+12\sqrt{249})}^{1/3}}{6}+\frac{4}{3{(188+12\sqrt{249})}^{1/3}}+\frac{2}{3}}\\ -\frac{((-{(188+12\sqrt{3}\sqrt{83})}^{2/3}-8{(188+12\sqrt{3}\sqrt{83})}^{1/3}+8)\cos \left(\frac{\sqrt{3}({(188+12\sqrt{3}\sqrt{83})}^{2/3}+8)\ln \left(x\right)}{12{(188+12\sqrt{3}\sqrt{83})}^{1/3}}\right)+\sqrt{3}({(188+12\sqrt{3}\sqrt{83})}^{2/3}+8)\sin \left(\frac{\sqrt{3}({(188+12\sqrt{3}\sqrt{83})}^{2/3}+8)\ln \left(x\right)}{12{(188+12\sqrt{3}\sqrt{83})}^{1/3}}\right))x^{\frac{{(188+12\sqrt{3}\sqrt{83})}^{2/3}-4{(188+12\sqrt{3}\sqrt{83})}^{1/3}-8}{12{(188+12\sqrt{3}\sqrt{83})}^{1/3}}}}{12{(188+12\sqrt{3}\sqrt{83})}^{1/3}}& -\frac{x^{-\frac{{(188+12\sqrt{249})}^{2/3}+2{(188+12\sqrt{249})}^{1/3}-8}{6{(188+12\sqrt{249})}^{1/3}}}({(188+12\sqrt{249})}^{2/3}-4{(188+12\sqrt{249})}^{1/3}-8)}{6{(188+12\sqrt{249})}^{1/3}}\end{array}\right]\\ & =-\frac{((i\sqrt{3}-3){(188+12\sqrt{249})}^{1/3}+\frac{47i\sqrt{3}}{2}+\frac{9i\sqrt{83}}{2}+\frac{9\sqrt{249}}{2}+\frac{141}{2})x^{\frac{2i\sqrt{3}{(188+12\sqrt{3}\sqrt{83})}^{1/3}+47i\sqrt{3}+9i\sqrt{83}-3\sqrt{3}\sqrt{83}+{(188+12\sqrt{3}\sqrt{83})}^{2/3}+2{(188+12\sqrt{3}\sqrt{83})}^{1/3}-47}{3{(188+12\sqrt{3}\sqrt{83})}^{2/3}}}-x^{\frac{(-2i\sqrt{3}+2){(188+12\sqrt{3}\sqrt{83})}^{1/3}+{(188+12\sqrt{3}\sqrt{83})}^{2/3}+(-9i-3\sqrt{3})\sqrt{83}-47i\sqrt{3}-47}{3{(188+12\sqrt{3}\sqrt{83})}^{2/3}}}((i\sqrt{3}+3){(188+12\sqrt{249})}^{1/3}+\frac{47i\sqrt{3}}{2}+\frac{9i\sqrt{83}}{2}-\frac{9\sqrt{249}}{2}-\frac{141}{2})}{3{(188+12\sqrt{249})}^{2/3}}\end{array}$$

$$\begin{array}{rl}{W}_3(x)& =det\left[\begin{array}{cc}{x}^{\frac{{(188+12\sqrt{249})}^{2/3}+8{(188+12\sqrt{249})}^{1/3}-8}{12{(188+12\sqrt{249})}^{1/3}}}\cos \left(\frac{\sqrt{3}({(188+12\sqrt{3}\sqrt{83})}^{2/3}+8)\ln \left(x\right)}{12{(188+12\sqrt{3}\sqrt{83})}^{1/3}}\right)& x^{\frac{{(188+12\sqrt{249})}^{2/3}+8{(188+12\sqrt{249})}^{1/3}-8}{12{(188+12\sqrt{249})}^{1/3}}}\sin \left(\frac{\sqrt{3}({(188+12\sqrt{3}\sqrt{83})}^{2/3}+8)\ln \left(x\right)}{12{(188+12\sqrt{3}\sqrt{83})}^{1/3}}\right)\\ -\frac{((-{(188+12\sqrt{3}\sqrt{83})}^{2/3}-8{(188+12\sqrt{3}\sqrt{83})}^{1/3}+8)\cos \left(\frac{\sqrt{3}({(188+12\sqrt{3}\sqrt{83})}^{2/3}+8)\ln \left(x\right)}{12{(188+12\sqrt{3}\sqrt{83})}^{1/3}}\right)+\sqrt{3}({(188+12\sqrt{3}\sqrt{83})}^{2/3}+8)\sin \left(\frac{\sqrt{3}({(188+12\sqrt{3}\sqrt{83})}^{2/3}+8)\ln \left(x\right)}{12{(188+12\sqrt{3}\sqrt{83})}^{1/3}}\right))x^{\frac{{(188+12\sqrt{3}\sqrt{83})}^{2/3}-4{(188+12\sqrt{3}\sqrt{83})}^{1/3}-8}{12{(188+12\sqrt{3}\sqrt{83})}^{1/3}}}}{12{(188+12\sqrt{3}\sqrt{83})}^{1/3}}& \frac{x^{\frac{{(188+12\sqrt{3}\sqrt{83})}^{2/3}-4{(188+12\sqrt{3}\sqrt{83})}^{1/3}-8}{12{(188+12\sqrt{3}\sqrt{83})}^{1/3}}}(({(188+12\sqrt{3}\sqrt{83})}^{2/3}+8{(188+12\sqrt{3}\sqrt{83})}^{1/3}-8)\sin \left(\frac{\sqrt{3}({(188+12\sqrt{3}\sqrt{83})}^{2/3}+8)\ln \left(x\right)}{12{(188+12\sqrt{3}\sqrt{83})}^{1/3}}\right)+\sqrt{3}({(188+12\sqrt{3}\sqrt{83})}^{2/3}+8)\cos \left(\frac{\sqrt{3}({(188+12\sqrt{3}\sqrt{83})}^{2/3}+8)\ln \left(x\right)}{12{(188+12\sqrt{3}\sqrt{83})}^{1/3}}\right))}{12{(188+12\sqrt{3}\sqrt{83})}^{1/3}}\end{array}\right]\\ & =\frac{\sqrt{3}({(188+12\sqrt{3}\sqrt{83})}^{2/3}+8)x^{\frac{{(188+12\sqrt{3}\sqrt{83})}^{2/3}+2{(188+12\sqrt{3}\sqrt{83})}^{1/3}-8}{6{(188+12\sqrt{3}\sqrt{83})}^{1/3}}}}{12{(188+12\sqrt{3}\sqrt{83})}^{1/3}}\end{array}$$

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

$$\begin{array}{rl}{U}_1& =(-1{)}^{3-1}\int \frac{F(x)W_1(x)}{aW(x)}dx\\ & =(-1{)}^2\int \frac{\left(x\right)\left(\frac{\frac{3((-\frac{2i}{3}-\frac{2\sqrt{3}}{9}){(188+12\sqrt{249})}^{1/3}+i\sqrt{249}+\frac{47i}{3}-\frac{47\sqrt{3}}{9}-\sqrt{83})x^{\frac{2i\sqrt{3}{(188+12\sqrt{3}\sqrt{83})}^{1/3}+47i\sqrt{3}+9i\sqrt{83}-3\sqrt{3}\sqrt{83}+{(188+12\sqrt{3}\sqrt{83})}^{2/3}+2{(188+12\sqrt{3}\sqrt{83})}^{1/3}-47}{3{(188+12\sqrt{3}\sqrt{83})}^{2/3}}}}{2}-\frac{3x^{\frac{(-2i\sqrt{3}+2){(188+12\sqrt{3}\sqrt{83})}^{1/3}+{(188+12\sqrt{3}\sqrt{83})}^{2/3}+(-9i-3\sqrt{3})\sqrt{83}-47i\sqrt{3}-47}{3{(188+12\sqrt{3}\sqrt{83})}^{2/3}}}((-\frac{2i}{3}+\frac{2\sqrt{3}}{9}){(188+12\sqrt{249})}^{1/3}+i\sqrt{249}+\frac{47i}{3}+\frac{47\sqrt{3}}{9}+\sqrt{83})}{2}}{{(188+12\sqrt{249})}^{2/3}}\right)}{\left(x^3\right)\left(\frac{\sqrt{83}}{2x}\right)}dx\\ & =\int \frac{\frac{3x(((-\frac{2i}{3}-\frac{2\sqrt{3}}{9}){(188+12\sqrt{249})}^{1/3}+i\sqrt{249}+\frac{47i}{3}-\frac{47\sqrt{3}}{9}-\sqrt{83})x^{\frac{2i\sqrt{3}{(188+12\sqrt{3}\sqrt{83})}^{1/3}+47i\sqrt{3}+9i\sqrt{83}-3\sqrt{3}\sqrt{83}+{(188+12\sqrt{3}\sqrt{83})}^{2/3}+2{(188+12\sqrt{3}\sqrt{83})}^{1/3}-47}{3{(188+12\sqrt{3}\sqrt{83})}^{2/3}}}-x^{\frac{(-2i\sqrt{3}+2){(188+12\sqrt{3}\sqrt{83})}^{1/3}+{(188+12\sqrt{3}\sqrt{83})}^{2/3}+(-9i-3\sqrt{3})\sqrt{83}-47i\sqrt{3}-47}{3{(188+12\sqrt{3}\sqrt{83})}^{2/3}}}((-\frac{2i}{3}+\frac{2\sqrt{3}}{9}){(188+12\sqrt{249})}^{1/3}+i\sqrt{249}+\frac{47i}{3}+\frac{47\sqrt{3}}{9}+\sqrt{83}))}{2{(188+12\sqrt{249})}^{2/3}}}{\frac{x^2\sqrt{83}}{2}}dx\\ & =\int \left(\frac{3(((-\frac{2i}{3}-\frac{2\sqrt{3}}{9}){(188+12\sqrt{249})}^{1/3}+i\sqrt{249}+\frac{47i}{3}-\frac{47\sqrt{3}}{9}-\sqrt{83})x^{\frac{2i\sqrt{3}{(188+12\sqrt{3}\sqrt{83})}^{1/3}+47i\sqrt{3}+9i\sqrt{83}-3\sqrt{3}\sqrt{83}+{(188+12\sqrt{3}\sqrt{83})}^{2/3}+2{(188+12\sqrt{3}\sqrt{83})}^{1/3}-47}{3{(188+12\sqrt{3}\sqrt{83})}^{2/3}}}-x^{\frac{(-2i\sqrt{3}+2){(188+12\sqrt{3}\sqrt{83})}^{1/3}+{(188+12\sqrt{3}\sqrt{83})}^{2/3}+(-9i-3\sqrt{3})\sqrt{83}-47i\sqrt{3}-47}{3{(188+12\sqrt{3}\sqrt{83})}^{2/3}}}((-\frac{2i}{3}+\frac{2\sqrt{3}}{9}){(188+12\sqrt{249})}^{1/3}+i\sqrt{249}+\frac{47i}{3}+\frac{47\sqrt{3}}{9}+\sqrt{83}))\sqrt{83}}{83x{(188+12\sqrt{249})}^{2/3}}\right)dx\\ & =\frac{(-\frac{3i\sqrt{83}}{332}+\frac{\sqrt{3}\sqrt{83}}{332}-\frac{{(188+12\sqrt{3}\sqrt{83})}^{2/3}}{96}-\frac{i{(188+12\sqrt{3}\sqrt{83})}^{2/3}\sqrt{3}}{96}+\frac{3i{(188+12\sqrt{3}\sqrt{83})}^{2/3}\sqrt{83}}{2656}+\frac{{(188+12\sqrt{3}\sqrt{83})}^{2/3}\sqrt{3}\sqrt{83}}{2656}+\frac{{(188+12\sqrt{3}\sqrt{83})}^{1/3}}{6}-\frac{i\sqrt{3}}{12}+\frac{1}{12}){\mathrm{e}}^{\frac{((-2i\sqrt{3}+2){(188+12\sqrt{3}\sqrt{83})}^{1/3}+{(188+12\sqrt{3}\sqrt{83})}^{2/3}+(-9i-3\sqrt{3})\sqrt{83}-47i\sqrt{3}-47)\ln \left(x\right)}{3{(188+12\sqrt{3}\sqrt{83})}^{2/3}}}+(\frac{3i\sqrt{83}}{332}+\frac{\sqrt{3}\sqrt{83}}{332}-\frac{{(188+12\sqrt{3}\sqrt{83})}^{2/3}}{96}+\frac{i{(188+12\sqrt{3}\sqrt{83})}^{2/3}\sqrt{3}}{96}-\frac{3i{(188+12\sqrt{3}\sqrt{83})}^{2/3}\sqrt{83}}{2656}+\frac{{(188+12\sqrt{3}\sqrt{83})}^{2/3}\sqrt{3}\sqrt{83}}{2656}+\frac{{(188+12\sqrt{3}\sqrt{83})}^{1/3}}{6}+\frac{i\sqrt{3}}{12}+\frac{1}{12}){\mathrm{e}}^{\frac{(2i\sqrt{3}{(188+12\sqrt{3}\sqrt{83})}^{1/3}+47i\sqrt{3}+9i\sqrt{83}-3\sqrt{3}\sqrt{83}+{(188+12\sqrt{3}\sqrt{83})}^{2/3}+2{(188+12\sqrt{3}\sqrt{83})}^{1/3}-47)\ln \left(x\right)}{3{(188+12\sqrt{3}\sqrt{83})}^{2/3}}}}{{(188+12\sqrt{3}\sqrt{83})}^{1/3}}\end{array}$$

$$\begin{array}{rl}{U}_2& =(-1{)}^{3-2}\int \frac{F(x)W_2(x)}{aW(x)}dx\\ & =(-1{)}^1\int \frac{\left(x\right)(-\frac{((i\sqrt{3}-3){(188+12\sqrt{249})}^{1/3}+\frac{47i\sqrt{3}}{2}+\frac{9i\sqrt{83}}{2}+\frac{9\sqrt{249}}{2}+\frac{141}{2})x^{\frac{2i\sqrt{3}{(188+12\sqrt{3}\sqrt{83})}^{1/3}+47i\sqrt{3}+9i\sqrt{83}-3\sqrt{3}\sqrt{83}+{(188+12\sqrt{3}\sqrt{83})}^{2/3}+2{(188+12\sqrt{3}\sqrt{83})}^{1/3}-47}{3{(188+12\sqrt{3}\sqrt{83})}^{2/3}}}-x^{\frac{(-2i\sqrt{3}+2){(188+12\sqrt{3}\sqrt{83})}^{1/3}+{(188+12\sqrt{3}\sqrt{83})}^{2/3}+(-9i-3\sqrt{3})\sqrt{83}-47i\sqrt{3}-47}{3{(188+12\sqrt{3}\sqrt{83})}^{2/3}}}((i\sqrt{3}+3){(188+12\sqrt{249})}^{1/3}+\frac{47i\sqrt{3}}{2}+\frac{9i\sqrt{83}}{2}-\frac{9\sqrt{249}}{2}-\frac{141}{2})}{3{(188+12\sqrt{249})}^{2/3}})}{\left(x^3\right)\left(\frac{\sqrt{83}}{2x}\right)}dx\\ & =-\int \frac{-\frac{x(((i\sqrt{3}-3){(188+12\sqrt{249})}^{1/3}+\frac{47i\sqrt{3}}{2}+\frac{9i\sqrt{83}}{2}+\frac{9\sqrt{249}}{2}+\frac{141}{2})x^{\frac{2i\sqrt{3}{(188+12\sqrt{3}\sqrt{83})}^{1/3}+47i\sqrt{3}+9i\sqrt{83}-3\sqrt{3}\sqrt{83}+{(188+12\sqrt{3}\sqrt{83})}^{2/3}+2{(188+12\sqrt{3}\sqrt{83})}^{1/3}-47}{3{(188+12\sqrt{3}\sqrt{83})}^{2/3}}}-x^{\frac{(-2i\sqrt{3}+2){(188+12\sqrt{3}\sqrt{83})}^{1/3}+{(188+12\sqrt{3}\sqrt{83})}^{2/3}+(-9i-3\sqrt{3})\sqrt{83}-47i\sqrt{3}-47}{3{(188+12\sqrt{3}\sqrt{83})}^{2/3}}}((i\sqrt{3}+3){(188+12\sqrt{249})}^{1/3}+\frac{47i\sqrt{3}}{2}+\frac{9i\sqrt{83}}{2}-\frac{9\sqrt{249}}{2}-\frac{141}{2}))}{3{(188+12\sqrt{249})}^{2/3}}}{\frac{x^2\sqrt{83}}{2}}dx\\ & =-\int (-\frac{2(((i\sqrt{3}-3){(188+12\sqrt{249})}^{1/3}+\frac{47i\sqrt{3}}{2}+\frac{9i\sqrt{83}}{2}+\frac{9\sqrt{249}}{2}+\frac{141}{2})x^{\frac{2i\sqrt{3}{(188+12\sqrt{3}\sqrt{83})}^{1/3}+47i\sqrt{3}+9i\sqrt{83}-3\sqrt{3}\sqrt{83}+{(188+12\sqrt{3}\sqrt{83})}^{2/3}+2{(188+12\sqrt{3}\sqrt{83})}^{1/3}-47}{3{(188+12\sqrt{3}\sqrt{83})}^{2/3}}}-x^{\frac{(-2i\sqrt{3}+2){(188+12\sqrt{3}\sqrt{83})}^{1/3}+{(188+12\sqrt{3}\sqrt{83})}^{2/3}+(-9i-3\sqrt{3})\sqrt{83}-47i\sqrt{3}-47}{3{(188+12\sqrt{3}\sqrt{83})}^{2/3}}}((i\sqrt{3}+3){(188+12\sqrt{249})}^{1/3}+\frac{47i\sqrt{3}}{2}+\frac{9i\sqrt{83}}{2}-\frac{9\sqrt{249}}{2}-\frac{141}{2}))\sqrt{83}}{249x{(188+12\sqrt{249})}^{2/3}})dx\\ & =-\frac{(-\frac{3\sqrt{83}}{332}-\frac{i\sqrt{3}\sqrt{83}}{332}-\frac{i{(188+12\sqrt{3}\sqrt{83})}^{1/3}}{6}-\frac{\sqrt{3}{(188+12\sqrt{3}\sqrt{83})}^{2/3}}{96}+\frac{i{(188+12\sqrt{3}\sqrt{83})}^{2/3}}{96}+\frac{3\sqrt{83}{(188+12\sqrt{3}\sqrt{83})}^{2/3}}{2656}-\frac{i{(188+12\sqrt{3}\sqrt{83})}^{2/3}\sqrt{3}\sqrt{83}}{2656}-\frac{i}{12}-\frac{\sqrt{3}}{12}){\mathrm{e}}^{\frac{((-2i\sqrt{3}+2){(188+12\sqrt{3}\sqrt{83})}^{1/3}+{(188+12\sqrt{3}\sqrt{83})}^{2/3}+(-9i-3\sqrt{3})\sqrt{83}-47i\sqrt{3}-47)\ln \left(x\right)}{3{(188+12\sqrt{3}\sqrt{83})}^{2/3}}}+(-\frac{3\sqrt{83}}{332}+\frac{i\sqrt{3}\sqrt{83}}{332}+\frac{i{(188+12\sqrt{3}\sqrt{83})}^{1/3}}{6}-\frac{\sqrt{3}{(188+12\sqrt{3}\sqrt{83})}^{2/3}}{96}-\frac{i{(188+12\sqrt{3}\sqrt{83})}^{2/3}}{96}+\frac{3\sqrt{83}{(188+12\sqrt{3}\sqrt{83})}^{2/3}}{2656}+\frac{i{(188+12\sqrt{3}\sqrt{83})}^{2/3}\sqrt{3}\sqrt{83}}{2656}+\frac{i}{12}-\frac{\sqrt{3}}{12}){\mathrm{e}}^{\frac{(2i\sqrt{3}{(188+12\sqrt{3}\sqrt{83})}^{1/3}+47i\sqrt{3}+9i\sqrt{83}-3\sqrt{3}\sqrt{83}+{(188+12\sqrt{3}\sqrt{83})}^{2/3}+2{(188+12\sqrt{3}\sqrt{83})}^{1/3}-47)\ln \left(x\right)}{3{(188+12\sqrt{3}\sqrt{83})}^{2/3}}}}{{(188+12\sqrt{3}\sqrt{83})}^{1/3}}\end{array}$$

$$\begin{array}{rl}{U}_3& =(-1{)}^{3-3}\int \frac{F(x)W_3(x)}{aW(x)}dx\\ & =(-1{)}^0\int \frac{\left(x\right)\left(\frac{\sqrt{3}({(188+12\sqrt{3}\sqrt{83})}^{2/3}+8)x^{\frac{{(188+12\sqrt{3}\sqrt{83})}^{2/3}+2{(188+12\sqrt{3}\sqrt{83})}^{1/3}-8}{6{(188+12\sqrt{3}\sqrt{83})}^{1/3}}}}{12{(188+12\sqrt{3}\sqrt{83})}^{1/3}}\right)}{\left(x^3\right)\left(\frac{\sqrt{83}}{2x}\right)}dx\\ & =\int \frac{\frac{x\sqrt{3}({(188+12\sqrt{3}\sqrt{83})}^{2/3}+8)x^{\frac{{(188+12\sqrt{3}\sqrt{83})}^{2/3}+2{(188+12\sqrt{3}\sqrt{83})}^{1/3}-8}{6{(188+12\sqrt{3}\sqrt{83})}^{1/3}}}}{12{(188+12\sqrt{3}\sqrt{83})}^{1/3}}}{\frac{x^2\sqrt{83}}{2}}dx\\ & =\int \left(\frac{x^{\frac{{(188+12\sqrt{249})}^{2/3}-4{(188+12\sqrt{249})}^{1/3}-8}{6{(188+12\sqrt{249})}^{1/3}}}({(188+12\sqrt{249})}^{2/3}+8)\sqrt{249}}{498{(188+12\sqrt{249})}^{1/3}}\right)dx\\ & =\frac{x^{1+\frac{{(188+12\sqrt{249})}^{2/3}-4{(188+12\sqrt{249})}^{1/3}-8}{6{(188+12\sqrt{249})}^{1/3}}}\sqrt{249}({(188+12\sqrt{249})}^{2/3}+8)(\sqrt{249}{(188+12\sqrt{249})}^{2/3}-13{(188+12\sqrt{249})}^{2/3}+4{(188+12\sqrt{249})}^{1/3}\sqrt{249}-68{(188+12\sqrt{249})}^{1/3}+32)}{95616{(188+12\sqrt{249})}^{1/3}}\\ & =\frac{x^{1+\frac{{(188+12\sqrt{249})}^{2/3}-4{(188+12\sqrt{249})}^{1/3}-8}{6{(188+12\sqrt{249})}^{1/3}}}\sqrt{249}({(188+12\sqrt{249})}^{2/3}+8)(\sqrt{249}{(188+12\sqrt{249})}^{2/3}-13{(188+12\sqrt{249})}^{2/3}+4{(188+12\sqrt{249})}^{1/3}\sqrt{249}-68{(188+12\sqrt{249})}^{1/3}+32)}{95616{(188+12\sqrt{249})}^{1/3}}\end{array}$$

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{array}{rl}{y}_p& =\left(\frac{(-\frac{3i\sqrt{83}}{332}+\frac{\sqrt{3}\sqrt{83}}{332}-\frac{{(188+12\sqrt{3}\sqrt{83})}^{2/3}}{96}-\frac{i{(188+12\sqrt{3}\sqrt{83})}^{2/3}\sqrt{3}}{96}+\frac{3i{(188+12\sqrt{3}\sqrt{83})}^{2/3}\sqrt{83}}{2656}+\frac{{(188+12\sqrt{3}\sqrt{83})}^{2/3}\sqrt{3}\sqrt{83}}{2656}+\frac{{(188+12\sqrt{3}\sqrt{83})}^{1/3}}{6}-\frac{i\sqrt{3}}{12}+\frac{1}{12}){\mathrm{e}}^{\frac{((-2i\sqrt{3}+2){(188+12\sqrt{3}\sqrt{83})}^{1/3}+{(188+12\sqrt{3}\sqrt{83})}^{2/3}+(-9i-3\sqrt{3})\sqrt{83}-47i\sqrt{3}-47)\ln \left(x\right)}{3{(188+12\sqrt{3}\sqrt{83})}^{2/3}}}+(\frac{3i\sqrt{83}}{332}+\frac{\sqrt{3}\sqrt{83}}{332}-\frac{{(188+12\sqrt{3}\sqrt{83})}^{2/3}}{96}+\frac{i{(188+12\sqrt{3}\sqrt{83})}^{2/3}\sqrt{3}}{96}-\frac{3i{(188+12\sqrt{3}\sqrt{83})}^{2/3}\sqrt{83}}{2656}+\frac{{(188+12\sqrt{3}\sqrt{83})}^{2/3}\sqrt{3}\sqrt{83}}{2656}+\frac{{(188+12\sqrt{3}\sqrt{83})}^{1/3}}{6}+\frac{i\sqrt{3}}{12}+\frac{1}{12}){\mathrm{e}}^{\frac{(2i\sqrt{3}{(188+12\sqrt{3}\sqrt{83})}^{1/3}+47i\sqrt{3}+9i\sqrt{83}-3\sqrt{3}\sqrt{83}+{(188+12\sqrt{3}\sqrt{83})}^{2/3}+2{(188+12\sqrt{3}\sqrt{83})}^{1/3}-47)\ln \left(x\right)}{3{(188+12\sqrt{3}\sqrt{83})}^{2/3}}}}{{(188+12\sqrt{3}\sqrt{83})}^{1/3}}\right)(x^{\frac{{(188+12\sqrt{249})}^{1/3}}{12}-\frac{2}{3{(188+12\sqrt{249})}^{1/3}}+\frac{2}{3}}\cos \left(\frac{\sqrt{3}(-\frac{{(188+12\sqrt{249})}^{1/3}}{6}-\frac{4}{3{(188+12\sqrt{249})}^{1/3}})\ln \left(x\right)}{2}\right))\\ & +(-\frac{(-\frac{3\sqrt{83}}{332}-\frac{i\sqrt{3}\sqrt{83}}{332}-\frac{i{(188+12\sqrt{3}\sqrt{83})}^{1/3}}{6}-\frac{\sqrt{3}{(188+12\sqrt{3}\sqrt{83})}^{2/3}}{96}+\frac{i{(188+12\sqrt{3}\sqrt{83})}^{2/3}}{96}+\frac{3\sqrt{83}{(188+12\sqrt{3}\sqrt{83})}^{2/3}}{2656}-\frac{i{(188+12\sqrt{3}\sqrt{83})}^{2/3}\sqrt{3}\sqrt{83}}{2656}-\frac{i}{12}-\frac{\sqrt{3}}{12}){\mathrm{e}}^{\frac{((-2i\sqrt{3}+2){(188+12\sqrt{3}\sqrt{83})}^{1/3}+{(188+12\sqrt{3}\sqrt{83})}^{2/3}+(-9i-3\sqrt{3})\sqrt{83}-47i\sqrt{3}-47)\ln \left(x\right)}{3{(188+12\sqrt{3}\sqrt{83})}^{2/3}}}+(-\frac{3\sqrt{83}}{332}+\frac{i\sqrt{3}\sqrt{83}}{332}+\frac{i{(188+12\sqrt{3}\sqrt{83})}^{1/3}}{6}-\frac{\sqrt{3}{(188+12\sqrt{3}\sqrt{83})}^{2/3}}{96}-\frac{i{(188+12\sqrt{3}\sqrt{83})}^{2/3}}{96}+\frac{3\sqrt{83}{(188+12\sqrt{3}\sqrt{83})}^{2/3}}{2656}+\frac{i{(188+12\sqrt{3}\sqrt{83})}^{2/3}\sqrt{3}\sqrt{83}}{2656}+\frac{i}{12}-\frac{\sqrt{3}}{12}){\mathrm{e}}^{\frac{(2i\sqrt{3}{(188+12\sqrt{3}\sqrt{83})}^{1/3}+47i\sqrt{3}+9i\sqrt{83}-3\sqrt{3}\sqrt{83}+{(188+12\sqrt{3}\sqrt{83})}^{2/3}+2{(188+12\sqrt{3}\sqrt{83})}^{1/3}-47)\ln \left(x\right)}{3{(188+12\sqrt{3}\sqrt{83})}^{2/3}}}}{{(188+12\sqrt{3}\sqrt{83})}^{1/3}})(-x^{\frac{{(188+12\sqrt{249})}^{1/3}}{12}-\frac{2}{3{(188+12\sqrt{249})}^{1/3}}+\frac{2}{3}}\sin \left(\frac{\sqrt{3}(-\frac{{(188+12\sqrt{249})}^{1/3}}{6}-\frac{4}{3{(188+12\sqrt{249})}^{1/3}})\ln \left(x\right)}{2}\right))\\ & +\left(\frac{x^{1+\frac{{(188+12\sqrt{249})}^{2/3}-4{(188+12\sqrt{249})}^{1/3}-8}{6{(188+12\sqrt{249})}^{1/3}}}\sqrt{249}({(188+12\sqrt{249})}^{2/3}+8)(\sqrt{249}{(188+12\sqrt{249})}^{2/3}-13{(188+12\sqrt{249})}^{2/3}+4{(188+12\sqrt{249})}^{1/3}\sqrt{249}-68{(188+12\sqrt{249})}^{1/3}+32)}{95616{(188+12\sqrt{249})}^{1/3}}\right)\left(x^{-\frac{{(188+12\sqrt{249})}^{1/3}}{6}+\frac{4}{3{(188+12\sqrt{249})}^{1/3}}+\frac{2}{3}}\right)\end{array}$$

Therefore the particular solution is

$$y_p=\frac{x}{2}$$

Therefore the general solution is

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 223

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

$$y=c_2{x}^{\frac{(-3\sqrt{249}+47){(188+12\sqrt{249})}^{2/3}}{192}+\frac{{(188+12\sqrt{249})}^{1/3}}{12}+\frac{2}{3}}\cos \left(\frac{{(188+12\sqrt{3}\sqrt{83})}^{1/3}\sqrt{3}(3{(188+12\sqrt{3}\sqrt{83})}^{1/3}\sqrt{3}\sqrt{83}-47{(188+12\sqrt{3}\sqrt{83})}^{1/3}+16)\ln \left(x\right)}{192}\right)+c_3{x}^{\frac{(-3\sqrt{249}+47){(188+12\sqrt{249})}^{2/3}}{192}+\frac{{(188+12\sqrt{249})}^{1/3}}{12}+\frac{2}{3}}\sin \left(\frac{{(188+12\sqrt{3}\sqrt{83})}^{1/3}\sqrt{3}(3{(188+12\sqrt{3}\sqrt{83})}^{1/3}\sqrt{3}\sqrt{83}-47{(188+12\sqrt{3}\sqrt{83})}^{1/3}+16)\ln \left(x\right)}{192}\right)+c_1{x}^{\frac{(3\sqrt{249}-47){(188+12\sqrt{249})}^{2/3}}{96}-\frac{{(188+12\sqrt{249})}^{1/3}}{6}+\frac{2}{3}}+1$$

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 step by step

$$\begin{array}{lll}& & \text{Let’s solve}\\ & & x^4\left(\frac{d}{dx}\frac{d^2}{d{x}^2}y\left(x\right)\right)+x^3\left(\frac{d}{dx}\frac{d}{dx}y\left(x\right)\right)+x^2\left(\frac{d}{dx}y\left(x\right)\right)+xy\left(x\right)=x\\ \bullet & & \text{Highest derivative means the order of the ODE is}3\\ & & \frac{d}{dx}\frac{d^2}{d{x}^2}y\left(x\right)\end{array}$$

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 82

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

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

✓ Sympy. Time used: 1.379 (sec). Leaf size: 233

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**4*Derivative(y(x), (x, 3)) + x**3*Derivative(y(x), (x, 2)) + x**2*Derivative(y(x), x) + x*y(x) - x,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

$$y\left(x\right)=\frac{C_1}{x^{-\frac{2}{3}-\frac{2\sqrt[3]{2}}{3\sqrt[3]{47+3\sqrt{249}}}+\frac{2^{\frac{2}{3}}\sqrt[3]{47+3\sqrt{249}}}{6}}}+C_2{x}^{-\frac{\sqrt[3]{2}}{3\sqrt[3]{47+3\sqrt{249}}}+\frac{2^{\frac{2}{3}}\sqrt[3]{47+3\sqrt{249}}}{12}+\frac{2}{3}}\sin \left(\frac{\sqrt[3]{2}\sqrt{3}(\frac{4}{\sqrt[3]{47+3\sqrt{249}}}+\sqrt[3]{2}\sqrt[3]{47+3\sqrt{249}})\log \left(x\right)}{12}\right)+C_3{x}^{-\frac{\sqrt[3]{2}}{3\sqrt[3]{47+3\sqrt{249}}}+\frac{2^{\frac{2}{3}}\sqrt[3]{47+3\sqrt{249}}}{12}+\frac{2}{3}}\cos \left(\frac{\sqrt[3]{2}\sqrt{3}(\frac{4}{\sqrt[3]{47+3\sqrt{249}}}+\sqrt[3]{2}\sqrt[3]{47+3\sqrt{249}})\log \left(x\right)}{12}\right)+1$$