2.3.18 Problem 18

Internal problem ID [10150]
Book : Own collection of miscellaneous problems
Section : section 3.0
Problem number : 18
Date solved : Thursday, November 27, 2025 at 10:21:53 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

Solve

\begin{align*} x^{4} y^{\prime \prime \prime }+x^{3} y^{\prime \prime }+y^{\prime } x^{2}+x y&=0 \\ \end{align*}

Solved as higher order Euler type ode

Time used: 0.138 (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

\[ x^{4} y^{\prime \prime \prime }+x^{3} y^{\prime \prime }+y^{\prime } x^{2}+x y = 0 \]

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

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

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

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}}}{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_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}} \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 ) \\ y_3 &= 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}} \\ \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 184

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)+y(x)*x = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \,x^{-\frac {\left (188+12 \sqrt {249}\right )^{{2}/{3}}-4 \left (188+12 \sqrt {249}\right )^{{1}/{3}}-8}{6 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}}+c_2 \,x^{\frac {-8+\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}} \sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+8\right ) \ln \left (x \right )}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}\right )+c_3 \,x^{\frac {-8+\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}} \cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+8\right ) \ln \left (x \right )}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}\right ) \]

Maple trace

Methods for third order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
checking if the LODE has constant coefficients 
checking if the LODE is of Euler type 
<- LODE of Euler type successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{4} \left (\frac {d^{3}}{d x^{3}}y \left (x \right )\right )+x^{3} \left (\frac {d^{2}}{d x^{2}}y \left (x \right )\right )+x^{2} \left (\frac {d}{d x}y \left (x \right )\right )+x y \left (x \right )=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 3 \\ {} & {} & \frac {d^{3}}{d x^{3}}y \left (x \right ) \\ \bullet & {} & \textrm {Isolate 3rd derivative}\hspace {3pt} \\ {} & {} & \frac {d^{3}}{d x^{3}}y \left (x \right )=-\frac {y \left (x \right )}{x^{3}}-\frac {\left (\frac {d^{2}}{d x^{2}}y \left (x \right )\right ) x +\frac {d}{d x}y \left (x \right )}{x^{2}} \\ \bullet & {} & \textrm {Group terms with}\hspace {3pt} y \left (x \right )\hspace {3pt}\textrm {on the lhs of the ODE and the rest on the rhs of the ODE; ODE is linear}\hspace {3pt} \\ {} & {} & \frac {d^{3}}{d x^{3}}y \left (x \right )+\frac {\frac {d^{2}}{d x^{2}}y \left (x \right )}{x}+\frac {\frac {d}{d x}y \left (x \right )}{x^{2}}+\frac {y \left (x \right )}{x^{3}}=0 \\ \bullet & {} & \textrm {Multiply by denominators of the ODE}\hspace {3pt} \\ {} & {} & \left (\frac {d^{3}}{d x^{3}}y \left (x \right )\right ) x^{3}+\left (\frac {d^{2}}{d x^{2}}y \left (x \right )\right ) x^{2}+\left (\frac {d}{d x}y \left (x \right )\right ) x +y \left (x \right )=0 \\ \bullet & {} & \textrm {Make a change of variables}\hspace {3pt} \\ {} & {} & t =\ln \left (x \right ) \\ \square & {} & \textrm {Substitute the change of variables back into the ODE}\hspace {3pt} \\ {} & \circ & \textrm {Calculate the}\hspace {3pt} \hspace {3pt}\textrm {1st}\hspace {3pt} \hspace {3pt}\textrm {derivative of}\hspace {3pt} \hspace {3pt}\textrm {y}\hspace {3pt} \hspace {3pt}\textrm {with respect to}\hspace {3pt} \hspace {3pt}\textrm {x}\hspace {3pt} \hspace {3pt}\textrm {, using the chain rule}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\left (\frac {d}{d t}y \left (t \right )\right ) \left (\frac {d}{d x}t \left (x \right )\right ) \\ {} & \circ & \textrm {Compute derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {\frac {d}{d t}y \left (t \right )}{x} \\ {} & \circ & \textrm {Calculate the}\hspace {3pt} \hspace {3pt}\textrm {2nd}\hspace {3pt} \hspace {3pt}\textrm {derivative of}\hspace {3pt} \hspace {3pt}\textrm {y}\hspace {3pt} \hspace {3pt}\textrm {with respect to}\hspace {3pt} \hspace {3pt}\textrm {x}\hspace {3pt} \hspace {3pt}\textrm {, using the chain rule}\hspace {3pt} \\ {} & {} & \frac {d^{2}}{d x^{2}}y \left (x \right )=\left (\frac {d^{2}}{d t^{2}}y \left (t \right )\right ) \left (\frac {d}{d x}t \left (x \right )\right )^{2}+\left (\frac {d^{2}}{d x^{2}}t \left (x \right )\right ) \left (\frac {d}{d t}y \left (t \right )\right ) \\ {} & \circ & \textrm {Compute derivative}\hspace {3pt} \\ {} & {} & \frac {d^{2}}{d x^{2}}y \left (x \right )=\frac {\frac {d^{2}}{d t^{2}}y \left (t \right )}{x^{2}}-\frac {\frac {d}{d t}y \left (t \right )}{x^{2}} \\ {} & \circ & \textrm {Calculate the}\hspace {3pt} \hspace {3pt}\textrm {3rd}\hspace {3pt} \hspace {3pt}\textrm {derivative of}\hspace {3pt} \hspace {3pt}\textrm {y}\hspace {3pt} \hspace {3pt}\textrm {with respect to}\hspace {3pt} \hspace {3pt}\textrm {x}\hspace {3pt} \hspace {3pt}\textrm {, using the chain rule}\hspace {3pt} \\ {} & {} & \frac {d^{3}}{d x^{3}}y \left (x \right )=\left (\frac {d^{3}}{d t^{3}}y \left (t \right )\right ) \left (\frac {d}{d x}t \left (x \right )\right )^{3}+3 \left (\frac {d}{d x}t \left (x \right )\right ) \left (\frac {d^{2}}{d x^{2}}t \left (x \right )\right ) \left (\frac {d^{2}}{d t^{2}}y \left (t \right )\right )+\left (\frac {d^{3}}{d x^{3}}t \left (x \right )\right ) \left (\frac {d}{d t}y \left (t \right )\right ) \\ {} & \circ & \textrm {Compute derivative}\hspace {3pt} \\ {} & {} & \frac {d^{3}}{d x^{3}}y \left (x \right )=\frac {\frac {d^{3}}{d t^{3}}y \left (t \right )}{x^{3}}-\frac {3 \left (\frac {d^{2}}{d t^{2}}y \left (t \right )\right )}{x^{3}}+\frac {2 \left (\frac {d}{d t}y \left (t \right )\right )}{x^{3}} \\ & {} & \textrm {Substitute the change of variables back into the ODE}\hspace {3pt} \\ {} & {} & \left (\frac {\frac {d^{3}}{d t^{3}}y \left (t \right )}{x^{3}}-\frac {3 \left (\frac {d^{2}}{d t^{2}}y \left (t \right )\right )}{x^{3}}+\frac {2 \left (\frac {d}{d t}y \left (t \right )\right )}{x^{3}}\right ) x^{3}+\left (\frac {\frac {d^{2}}{d t^{2}}y \left (t \right )}{x^{2}}-\frac {\frac {d}{d t}y \left (t \right )}{x^{2}}\right ) x^{2}+\frac {d}{d t}y \left (t \right )+y \left (t \right )=0 \\ \bullet & {} & \textrm {Simplify}\hspace {3pt} \\ {} & {} & \frac {d^{3}}{d t^{3}}y \left (t \right )-2 \frac {d^{2}}{d t^{2}}y \left (t \right )+2 \frac {d}{d t}y \left (t \right )+y \left (t \right )=0 \\ \bullet & {} & \textrm {Characteristic polynomial of ODE}\hspace {3pt} \\ {} & {} & r^{3}-2 r^{2}+2 r +1=0 \\ \bullet & {} & \textrm {Roots of the characteristic polynomial}\hspace {3pt} \\ {} & {} & r =\left [-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}+\frac {4}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{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 {\mathrm {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}, \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 {\mathrm {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}\right ] \\ \bullet & {} & \textrm {Solution from}\hspace {3pt} r =-\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_{1}\left (t \right )={\mathrm e}^{\left (-\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 ) t} \\ \bullet & {} & \textrm {Solutions from}\hspace {3pt} r =\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 {\mathrm {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}\hspace {3pt}\textrm {and}\hspace {3pt} r =\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 {\mathrm {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} \\ {} & {} & \left [y_{2}\left (t \right )=-{\mathrm e}^{\left (\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}\right ) t} \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 ) t}{2}\right ), y_{3}\left (t \right )={\mathrm e}^{\left (\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{12}-\frac {2}{3 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}+\frac {2}{3}\right ) t} \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 ) t}{2}\right )\right ] \\ \bullet & {} & \textrm {General solution of the ODE}\hspace {3pt} \\ {} & {} & y \left (t \right )=\mathit {C1} y_{1}\left (t \right )+\mathit {C2} y_{2}\left (t \right )+\mathit {C3} y_{3}\left (t \right ) \\ \bullet & {} & \textrm {Substitute in solutions and simplify}\hspace {3pt} \\ {} & {} & y \left (t \right )=\mathit {C1} \,{\mathrm e}^{-\frac {\left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}-4 \left (188+12 \sqrt {249}\right )^{{1}/{3}}-8\right ) t}{6 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}}+{\mathrm e}^{\frac {\left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8 \left (188+12 \sqrt {249}\right )^{{1}/{3}}-8\right ) t}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}} \left (\sin \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}\right ) \mathit {C2} +\cos \left (\frac {\sqrt {3}\, \left (\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{2}/{3}}+8\right ) t}{12 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}}\right ) \mathit {C3} \right ) \\ \bullet & {} & \textrm {Change variables back using}\hspace {3pt} t =\ln \left (x \right ) \\ {} & {} & y \left (x \right )=\mathit {C1} \,{\mathrm e}^{-\frac {\left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}-4 \left (188+12 \sqrt {249}\right )^{{1}/{3}}-8\right ) \ln \left (x \right )}{6 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}}+{\mathrm e}^{\frac {\left (\left (188+12 \sqrt {249}\right )^{{2}/{3}}+8 \left (188+12 \sqrt {249}\right )^{{1}/{3}}-8\right ) \ln \left (x \right )}{12 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}} \left (\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 ) \mathit {C2} +\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 ) \mathit {C3} \right ) \\ \bullet & {} & \textrm {Simplify}\hspace {3pt} \\ {} & {} & y \left (x \right )=x^{{2}/{3}} \left (\mathit {C2} \,x^{\frac {\left (188+12 \sqrt {249}\right )^{{2}/{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 )+\mathit {C3} \,x^{\frac {\left (188+12 \sqrt {249}\right )^{{2}/{3}}-8}{12 \left (188+12 \sqrt {249}\right )^{{1}/{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 )+\mathit {C1} \,x^{-\frac {\left (188+12 \sqrt {249}\right )^{{2}/{3}}-8}{6 \left (188+12 \sqrt {249}\right )^{{1}/{3}}}}\right ) \end {array} \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 81

ode=x^4*D[y[x],{x,3}]+x^3*D[y[x],{x,2}]+x^2*D[y[x],x]+x*y[x]== 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} 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 ]} \end{align*}

✓ Sympy. Time used: 0.142 (sec). Leaf size: 54

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),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {C_{1}}{x^{- \operatorname {CRootOf} {\left (x^{3} - 2 x^{2} + 2 x + 1, 0\right )}}} + C_{2} x^{\operatorname {CRootOf} {\left (x^{3} - 2 x^{2} + 2 x + 1, 1\right )}} + C_{3} x^{\operatorname {CRootOf} {\left (x^{3} - 2 x^{2} + 2 x + 1, 2\right )}} \]