2.3.20 Problem 20

Internal problem ID [10152]
Book : Own collection of miscellaneous problems
Section : section 3.0
Problem number : 20
Date solved : Monday, December 08, 2025 at 07:43:09 PM
CAS classification : [[_high_order, _with_linear_symmetries]]

Entering higher order ode solver

\begin{align*} 5 x^{5} y^{\prime \prime \prime \prime }+4 x^{4} y^{\prime \prime \prime }+y^{\prime } x^{2}+x y&=0 \\ \end{align*}

2.3.20.1 Solved as higher order Euler type ode

0.296 (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}\\ y^{\prime \prime \prime \prime } &= \lambda \left (\lambda -1\right ) \left (\lambda -2\right ) \left (\lambda -3\right ) x^{\lambda -4} \end{align*}

Substituting these back into

\[ 5 x^{5} y^{\prime \prime \prime \prime }+4 x^{4} y^{\prime \prime \prime }+y^{\prime } x^{2}+x y = 0 \]

gives

\[ x \lambda \,x^{\lambda -1}+4 x^{3} \lambda \left (\lambda -1\right ) \left (\lambda -2\right ) x^{\lambda -3}+5 x^{4} \lambda \left (\lambda -1\right ) \left (\lambda -2\right ) \left (\lambda -3\right ) x^{\lambda -4}+x^{\lambda } = 0 \]

Which simplifies to

\[ \lambda \,x^{\lambda }+4 \lambda \left (\lambda -1\right ) \left (\lambda -2\right ) x^{\lambda }+5 \lambda \left (\lambda -1\right ) \left (\lambda -2\right ) \left (\lambda -3\right ) x^{\lambda }+x^{\lambda } = 0 \]

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

\[ \lambda +4 \lambda \left (\lambda -1\right ) \left (\lambda -2\right )+5 \lambda \left (\lambda -1\right ) \left (\lambda -2\right ) \left (\lambda -3\right )+1 = 0 \]

Simplifying gives the characteristic equation as

\[ 5 \lambda ^{4}-26 \lambda ^{3}+43 \lambda ^{2}-21 \lambda +1 = 0 \]

Solving the above gives the following roots

\begin{align*} \lambda _1 &= \frac {13}{10}+\frac {\sqrt {6}\, \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}}{60}+\frac {i \sqrt {6}\, \sqrt {\frac {5 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}\, \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+876 \sqrt {6}\, \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}-308 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}+5420 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}}}}{60}\\ \lambda _2 &= \frac {13}{10}+\frac {\sqrt {6}\, \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}}{60}-\frac {i \sqrt {6}\, \sqrt {\frac {5 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}\, \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+876 \sqrt {6}\, \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}-308 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}+5420 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}}}}{60}\\ \lambda _3 &= \frac {13}{10}-\frac {\sqrt {6}\, \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}}{60}+\frac {\sqrt {6}\, \sqrt {\frac {-5 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}\, \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+876 \sqrt {6}\, \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+308 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}-5420 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}}}}{60}\\ \lambda _4 &= \frac {13}{10}-\frac {\sqrt {6}\, \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}}{60}-\frac {\sqrt {6}\, \sqrt {\frac {-5 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}\, \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+876 \sqrt {6}\, \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+308 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}-5420 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}}}}{60} \end{align*}

This table summarises the result

root	multiplicity	type of root
\(\frac {13}{10}-\frac {\sqrt {6}\, \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}}{60}+\frac {\sqrt {6}\, \sqrt {\frac {-5 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}\, \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+876 \sqrt {6}\, \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+308 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}-5420 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}}}}{60}\)	\(1\)	real root
\(\frac {13}{10}-\frac {\sqrt {6}\, \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}}{60}-\frac {\sqrt {6}\, \sqrt {\frac {-5 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}\, \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+876 \sqrt {6}\, \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+308 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}-5420 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}}}}{60}\)	\(1\)	real root
\(\frac {13}{10}+\frac {\sqrt {6}\, \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}}{60} \pm \frac {\sqrt {6}\, \sqrt {\frac {5 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}\, \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+876 \sqrt {6}\, \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}-308 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}+5420 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}}}}{60} i\)	\(1\)	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

\[ \text {Expression too large to display} \]

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

\begin{align*} y_1 &= x^{\frac {13}{10}-\frac {\sqrt {6}\, \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}}{60}+\frac {\sqrt {6}\, \sqrt {\frac {-5 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}\, \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+876 \sqrt {6}\, \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+308 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}-5420 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}}}}{60}} \\ y_2 &= x^{\frac {13}{10}-\frac {\sqrt {6}\, \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}}{60}-\frac {\sqrt {6}\, \sqrt {\frac {-5 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}\, \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+876 \sqrt {6}\, \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+308 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}-5420 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}}}}{60}} \\ y_3 &= x^{\frac {13}{10}+\frac {\sqrt {6}\, \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}}{60}} \cos \left (\frac {\sqrt {6}\, \sqrt {\frac {5 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}\, \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+876 \sqrt {6}\, \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}-308 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}+5420 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}}}\, \ln \left (x \right )}{60}\right ) \\ y_4 &= x^{\frac {13}{10}+\frac {\sqrt {6}\, \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}}{60}} \sin \left (\frac {\sqrt {6}\, \sqrt {\frac {5 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}\, \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+876 \sqrt {6}\, \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}-308 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}+5420 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}}}\, \ln \left (x \right )}{60}\right ) \\ \end{align*}

2.3.20.2 ✓ Maple. Time used: 0.002 (sec). Leaf size: 38

ode:=5*x^5*diff(diff(diff(diff(y(x),x),x),x),x)+4*x^4*diff(diff(diff(y(x),x),x),x)+x^2*diff(y(x),x)+y(x)*x = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \moverset {4}{\munderset {\textit {\_a} =1}{\sum }}x^{\operatorname {RootOf}\left (5 \textit {\_Z}^{4}-26 \textit {\_Z}^{3}+43 \textit {\_Z}^{2}-21 \textit {\_Z} +1, \operatorname {index} =\textit {\_a} \right )} \textit {\_C}_{\textit {\_a}} \]

Maple trace

Methods for high 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} \\ {} & {} & 5 x^{5} \left (\frac {d^{4}}{d x^{4}}y \left (x \right )\right )+4 x^{4} \left (\frac {d^{3}}{d x^{3}}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} 4 \\ {} & {} & \frac {d^{4}}{d x^{4}}y \left (x \right ) \\ \bullet & {} & \textrm {Isolate 4th derivative}\hspace {3pt} \\ {} & {} & \frac {d^{4}}{d x^{4}}y \left (x \right )=-\frac {y \left (x \right )}{5 x^{4}}-\frac {4 \left (\frac {d^{3}}{d x^{3}}y \left (x \right )\right ) x^{2}+\frac {d}{d x}y \left (x \right )}{5 x^{3}} \\ \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^{4}}{d x^{4}}y \left (x \right )+\frac {4 \left (\frac {d^{3}}{d x^{3}}y \left (x \right )\right )}{5 x}+\frac {\frac {d}{d x}y \left (x \right )}{5 x^{3}}+\frac {y \left (x \right )}{5 x^{4}}=0 \\ \bullet & {} & \textrm {Multiply by denominators of the ODE}\hspace {3pt} \\ {} & {} & 5 \left (\frac {d^{4}}{d x^{4}}y \left (x \right )\right ) x^{4}+4 \left (\frac {d^{3}}{d x^{3}}y \left (x \right )\right ) x^{3}+\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}} \\ {} & \circ & \textrm {Calculate the}\hspace {3pt} \hspace {3pt}\textrm {4th}\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^{4}}{d x^{4}}y \left (x \right )=\left (\frac {d^{4}}{d t^{4}}y \left (t \right )\right ) \left (\frac {d}{d x}t \left (x \right )\right )^{4}+3 \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^{3}}{d t^{3}}y \left (t \right )\right )+3 \left (\frac {d^{2}}{d x^{2}}t \left (x \right )\right )^{2} \left (\frac {d^{2}}{d t^{2}}y \left (t \right )\right )+3 \left (\left (\frac {d^{3}}{d x^{3}}t \left (x \right )\right ) \left (\frac {d^{2}}{d t^{2}}y \left (t \right )\right )+\left (\frac {d^{3}}{d t^{3}}y \left (t \right )\right ) \left (\frac {d}{d x}t \left (x \right )\right ) \left (\frac {d^{2}}{d x^{2}}t \left (x \right )\right )\right ) \left (\frac {d}{d x}t \left (x \right )\right )+\left (\frac {d^{4}}{d x^{4}}t \left (x \right )\right ) \left (\frac {d}{d t}y \left (t \right )\right )+\left (\frac {d^{2}}{d t^{2}}y \left (t \right )\right ) \left (\frac {d}{d x}t \left (x \right )\right ) \left (\frac {d^{3}}{d x^{3}}t \left (x \right )\right ) \\ {} & \circ & \textrm {Compute derivative}\hspace {3pt} \\ {} & {} & \frac {d^{4}}{d x^{4}}y \left (x \right )=\frac {\frac {d^{4}}{d t^{4}}y \left (t \right )}{x^{4}}-\frac {3 \left (\frac {d^{3}}{d t^{3}}y \left (t \right )\right )}{x^{4}}+\frac {5 \left (\frac {d^{2}}{d t^{2}}y \left (t \right )\right )}{x^{4}}+\frac {3 \left (\frac {2 \left (\frac {d^{2}}{d t^{2}}y \left (t \right )\right )}{x^{3}}-\frac {\frac {d^{3}}{d t^{3}}y \left (t \right )}{x^{3}}\right )}{x}-\frac {6 \left (\frac {d}{d t}y \left (t \right )\right )}{x^{4}} \\ & {} & \textrm {Substitute the change of variables back into the ODE}\hspace {3pt} \\ {} & {} & 5 \left (\frac {\frac {d^{4}}{d t^{4}}y \left (t \right )}{x^{4}}-\frac {3 \left (\frac {d^{3}}{d t^{3}}y \left (t \right )\right )}{x^{4}}+\frac {5 \left (\frac {d^{2}}{d t^{2}}y \left (t \right )\right )}{x^{4}}+\frac {3 \left (\frac {2 \left (\frac {d^{2}}{d t^{2}}y \left (t \right )\right )}{x^{3}}-\frac {\frac {d^{3}}{d t^{3}}y \left (t \right )}{x^{3}}\right )}{x}-\frac {6 \left (\frac {d}{d t}y \left (t \right )\right )}{x^{4}}\right ) x^{4}+4 \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}+\frac {d}{d t}y \left (t \right )+y \left (t \right )=0 \\ \bullet & {} & \textrm {Simplify}\hspace {3pt} \\ {} & {} & 5 \frac {d^{4}}{d t^{4}}y \left (t \right )-26 \frac {d^{3}}{d t^{3}}y \left (t \right )+43 \frac {d^{2}}{d t^{2}}y \left (t \right )-21 \frac {d}{d t}y \left (t \right )+y \left (t \right )=0 \\ \bullet & {} & \textrm {Characteristic polynomial of ODE}\hspace {3pt} \\ {} & {} & 5 r^{4}-26 r^{3}+43 r^{2}-21 r +1=0 \\ \bullet & {} & \textrm {Roots of the characteristic polynomial}\hspace {3pt} \\ {} & {} & r =\left [\mathit {RootOf}\left (5 \textit {\_Z}^{4}-26 \textit {\_Z}^{3}+43 \textit {\_Z}^{2}-21 \textit {\_Z} +1, \mathit {index} =1\right ), \mathit {RootOf}\left (5 \textit {\_Z}^{4}-26 \textit {\_Z}^{3}+43 \textit {\_Z}^{2}-21 \textit {\_Z} +1, \mathit {index} =2\right ), \mathit {RootOf}\left (5 \textit {\_Z}^{4}-26 \textit {\_Z}^{3}+43 \textit {\_Z}^{2}-21 \textit {\_Z} +1, \mathit {index} =3\right ), \mathit {RootOf}\left (5 \textit {\_Z}^{4}-26 \textit {\_Z}^{3}+43 \textit {\_Z}^{2}-21 \textit {\_Z} +1, \mathit {index} =4\right )\right ] \\ \bullet & {} & \textrm {Solution from}\hspace {3pt} r =\mathit {RootOf}\left (5 \textit {\_Z}^{4}-26 \textit {\_Z}^{3}+43 \textit {\_Z}^{2}-21 \textit {\_Z} +1, \mathit {index} =1\right ) \\ {} & {} & y_{1}\left (t \right )={\mathrm e}^{\mathit {RootOf}\left (5 \textit {\_Z}^{4}-26 \textit {\_Z}^{3}+43 \textit {\_Z}^{2}-21 \textit {\_Z} +1, \mathit {index} =1\right ) t} \\ \bullet & {} & \textrm {Solution from}\hspace {3pt} r =\mathit {RootOf}\left (5 \textit {\_Z}^{4}-26 \textit {\_Z}^{3}+43 \textit {\_Z}^{2}-21 \textit {\_Z} +1, \mathit {index} =2\right ) \\ {} & {} & y_{2}\left (t \right )={\mathrm e}^{\mathit {RootOf}\left (5 \textit {\_Z}^{4}-26 \textit {\_Z}^{3}+43 \textit {\_Z}^{2}-21 \textit {\_Z} +1, \mathit {index} =2\right ) t} \\ \bullet & {} & \textrm {Solutions from}\hspace {3pt} r =\mathit {RootOf}\left (5 \textit {\_Z}^{4}-26 \textit {\_Z}^{3}+43 \textit {\_Z}^{2}-21 \textit {\_Z} +1, \mathit {index} =3\right )\hspace {3pt}\textrm {and}\hspace {3pt} r =\mathit {RootOf}\left (5 \textit {\_Z}^{4}-26 \textit {\_Z}^{3}+43 \textit {\_Z}^{2}-21 \textit {\_Z} +1, \mathit {index} =4\right ) \\ {} & {} & \left [y_{3}\left (t \right )={\mathrm e}^{\left (\frac {13}{10}+\frac {\sqrt {6}\, \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}}{60}\right ) t} \sin \left (\frac {\sqrt {6}\, \sqrt {\frac {5 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}\, \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+876 \sqrt {6}\, \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}-308 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}+5420 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}}}\, t}{60}\right ), y_{4}\left (t \right )={\mathrm e}^{\left (\frac {13}{10}+\frac {\sqrt {6}\, \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}}{60}\right ) t} \cos \left (\frac {\sqrt {6}\, \sqrt {\frac {5 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}\, \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+876 \sqrt {6}\, \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}-308 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}+5420 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}}}\, t}{60}\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 )+\mathit {C4} y_{4}\left (t \right ) \\ \bullet & {} & \textrm {Substitute in solutions and simplify}\hspace {3pt} \\ {} & {} & y \left (t \right )=\mathit {C4} \,{\mathrm e}^{\frac {\left (\sqrt {3}\, 2^{{1}/{6}} \sqrt {10 \,2^{{1}/{3}} \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{2}/{3}}+154 \,2^{{2}/{3}} \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{1}/{3}}+5420}+78 \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{1}/{6}}\right ) t}{60 \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{1}/{6}}}} \cos \left (\frac {\sqrt {3}\, 2^{{5}/{6}} \sqrt {\left (-154 \,2^{{1}/{3}} \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{1}/{3}}+1355 \,2^{{2}/{3}}+5 \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{2}/{3}}\right ) \sqrt {10 \,2^{{1}/{3}} \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{2}/{3}}+154 \,2^{{2}/{3}} \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{1}/{3}}+5420}+876 \sqrt {3}\, 2^{{1}/{6}} \sqrt {3 \sqrt {3}\, \sqrt {769271}+10019}}\, t}{60 \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{1}/{6}} \left (10 \,2^{{1}/{3}} \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{2}/{3}}+154 \,2^{{2}/{3}} \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{1}/{3}}+5420\right )^{{1}/{4}}}\right )+\mathit {C3} \,{\mathrm e}^{\frac {\left (\sqrt {3}\, 2^{{1}/{6}} \sqrt {10 \,2^{{1}/{3}} \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{2}/{3}}+154 \,2^{{2}/{3}} \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{1}/{3}}+5420}+78 \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{1}/{6}}\right ) t}{60 \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{1}/{6}}}} \sin \left (\frac {\sqrt {3}\, 2^{{5}/{6}} \sqrt {\left (-154 \,2^{{1}/{3}} \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{1}/{3}}+1355 \,2^{{2}/{3}}+5 \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{2}/{3}}\right ) \sqrt {10 \,2^{{1}/{3}} \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{2}/{3}}+154 \,2^{{2}/{3}} \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{1}/{3}}+5420}+876 \sqrt {3}\, 2^{{1}/{6}} \sqrt {3 \sqrt {3}\, \sqrt {769271}+10019}}\, t}{60 \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{1}/{6}} \left (10 \,2^{{1}/{3}} \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{2}/{3}}+154 \,2^{{2}/{3}} \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{1}/{3}}+5420\right )^{{1}/{4}}}\right )+\mathit {C1} \,{\mathrm e}^{\mathit {RootOf}\left (5 \textit {\_Z}^{4}-26 \textit {\_Z}^{3}+43 \textit {\_Z}^{2}-21 \textit {\_Z} +1, \mathit {index} =1\right ) t}+\mathit {C2} \,{\mathrm e}^{\mathit {RootOf}\left (5 \textit {\_Z}^{4}-26 \textit {\_Z}^{3}+43 \textit {\_Z}^{2}-21 \textit {\_Z} +1, \mathit {index} =2\right ) t} \\ \bullet & {} & \textrm {Change variables back using}\hspace {3pt} t =\ln \left (x \right ) \\ {} & {} & y \left (x \right )=\mathit {C4} \,{\mathrm e}^{\frac {\left (\sqrt {3}\, 2^{{1}/{6}} \sqrt {10 \,2^{{1}/{3}} \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{2}/{3}}+154 \,2^{{2}/{3}} \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{1}/{3}}+5420}+78 \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{1}/{6}}\right ) \ln \left (x \right )}{60 \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{1}/{6}}}} \cos \left (\frac {\sqrt {3}\, 2^{{5}/{6}} \sqrt {\left (-154 \,2^{{1}/{3}} \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{1}/{3}}+1355 \,2^{{2}/{3}}+5 \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{2}/{3}}\right ) \sqrt {10 \,2^{{1}/{3}} \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{2}/{3}}+154 \,2^{{2}/{3}} \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{1}/{3}}+5420}+876 \sqrt {3}\, 2^{{1}/{6}} \sqrt {3 \sqrt {3}\, \sqrt {769271}+10019}}\, \ln \left (x \right )}{60 \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{1}/{6}} \left (10 \,2^{{1}/{3}} \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{2}/{3}}+154 \,2^{{2}/{3}} \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{1}/{3}}+5420\right )^{{1}/{4}}}\right )+\mathit {C3} \,{\mathrm e}^{\frac {\left (\sqrt {3}\, 2^{{1}/{6}} \sqrt {10 \,2^{{1}/{3}} \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{2}/{3}}+154 \,2^{{2}/{3}} \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{1}/{3}}+5420}+78 \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{1}/{6}}\right ) \ln \left (x \right )}{60 \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{1}/{6}}}} \sin \left (\frac {\sqrt {3}\, 2^{{5}/{6}} \sqrt {\left (-154 \,2^{{1}/{3}} \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{1}/{3}}+1355 \,2^{{2}/{3}}+5 \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{2}/{3}}\right ) \sqrt {10 \,2^{{1}/{3}} \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{2}/{3}}+154 \,2^{{2}/{3}} \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{1}/{3}}+5420}+876 \sqrt {3}\, 2^{{1}/{6}} \sqrt {3 \sqrt {3}\, \sqrt {769271}+10019}}\, \ln \left (x \right )}{60 \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{1}/{6}} \left (10 \,2^{{1}/{3}} \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{2}/{3}}+154 \,2^{{2}/{3}} \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{1}/{3}}+5420\right )^{{1}/{4}}}\right )+\mathit {C1} \,{\mathrm e}^{\mathit {RootOf}\left (5 \textit {\_Z}^{4}-26 \textit {\_Z}^{3}+43 \textit {\_Z}^{2}-21 \textit {\_Z} +1, \mathit {index} =1\right ) \ln \left (x \right )}+\mathit {C2} \,{\mathrm e}^{\mathit {RootOf}\left (5 \textit {\_Z}^{4}-26 \textit {\_Z}^{3}+43 \textit {\_Z}^{2}-21 \textit {\_Z} +1, \mathit {index} =2\right ) \ln \left (x \right )} \\ \bullet & {} & \textrm {Simplify}\hspace {3pt} \\ {} & {} & y \left (x \right )=\mathit {C4} \,x^{\frac {\sqrt {3}\, 2^{{1}/{6}} \sqrt {10 \,2^{{1}/{3}} \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{2}/{3}}+154 \,2^{{2}/{3}} \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{1}/{3}}+5420}+78 \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{1}/{6}}}{60 \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{1}/{6}}}} \cos \left (\frac {\sqrt {3}\, 2^{{5}/{6}} \sqrt {\left (-154 \,2^{{1}/{3}} \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{1}/{3}}+1355 \,2^{{2}/{3}}+5 \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{2}/{3}}\right ) \sqrt {10 \,2^{{1}/{3}} \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{2}/{3}}+154 \,2^{{2}/{3}} \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{1}/{3}}+5420}+876 \sqrt {3}\, 2^{{1}/{6}} \sqrt {3 \sqrt {3}\, \sqrt {769271}+10019}}\, \ln \left (x \right )}{60 \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{1}/{6}} \left (10 \,2^{{1}/{3}} \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{2}/{3}}+154 \,2^{{2}/{3}} \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{1}/{3}}+5420\right )^{{1}/{4}}}\right )+\mathit {C3} \,x^{\frac {\sqrt {3}\, 2^{{1}/{6}} \sqrt {10 \,2^{{1}/{3}} \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{2}/{3}}+154 \,2^{{2}/{3}} \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{1}/{3}}+5420}+78 \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{1}/{6}}}{60 \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{1}/{6}}}} \sin \left (\frac {\sqrt {3}\, 2^{{5}/{6}} \sqrt {\left (-154 \,2^{{1}/{3}} \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{1}/{3}}+1355 \,2^{{2}/{3}}+5 \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{2}/{3}}\right ) \sqrt {10 \,2^{{1}/{3}} \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{2}/{3}}+154 \,2^{{2}/{3}} \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{1}/{3}}+5420}+876 \sqrt {3}\, 2^{{1}/{6}} \sqrt {3 \sqrt {3}\, \sqrt {769271}+10019}}\, \ln \left (x \right )}{60 \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{1}/{6}} \left (10 \,2^{{1}/{3}} \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{2}/{3}}+154 \,2^{{2}/{3}} \left (3 \sqrt {3}\, \sqrt {769271}+10019\right )^{{1}/{3}}+5420\right )^{{1}/{4}}}\right )+\mathit {C1} \,x^{\mathit {RootOf}\left (5 \textit {\_Z}^{4}-26 \textit {\_Z}^{3}+43 \textit {\_Z}^{2}-21 \textit {\_Z} +1, \mathit {index} =1\right )}+\mathit {C2} \,x^{\mathit {RootOf}\left (5 \textit {\_Z}^{4}-26 \textit {\_Z}^{3}+43 \textit {\_Z}^{2}-21 \textit {\_Z} +1, \mathit {index} =2\right )} \end {array} \]

2.3.20.3 ✓ Mathematica. Time used: 1.033 (sec). Leaf size: 1880

ode=5*x^5*D[y[x],{x,4}]+4*x^4*D[y[x],{x,3}]+x^2*D[y[x],x]+x*y[x]== Sin[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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2.3.20.4 ✓ Sympy. Time used: 0.208 (sec). Leaf size: 97

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(5*x**5*Derivative(y(x), (x, 4)) + 4*x**4*Derivative(y(x), (x, 3)) + x**2*Derivative(y(x), x) + x*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = C_{1} x^{\operatorname {CRootOf} {\left (5 x^{4} - 26 x^{3} + 43 x^{2} - 21 x + 1, 0\right )}} + C_{2} x^{\operatorname {CRootOf} {\left (5 x^{4} - 26 x^{3} + 43 x^{2} - 21 x + 1, 1\right )}} + C_{3} x^{\operatorname {CRootOf} {\left (5 x^{4} - 26 x^{3} + 43 x^{2} - 21 x + 1, 2\right )}} + C_{4} x^{\operatorname {CRootOf} {\left (5 x^{4} - 26 x^{3} + 43 x^{2} - 21 x + 1, 3\right )}} \]