2.3.20 problem 20
Internal
problem
ID
[8304]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
3.0
Problem
number
:
20
Date
solved
:
Sunday, November 10, 2024 at 03:37:18 AM
CAS
classification
:
[[_high_order, _with_linear_symmetries]]
Solve
\begin{align*} 5 x^{5} y^{\prime \prime \prime \prime }+4 x^{4} y^{\prime \prime \prime }+x^{2} y^{\prime }+x y&=0 \end{align*}
The ODE is
\[ 5 x^{5} y^{\prime \prime \prime \prime }+4 x^{4} y^{\prime \prime \prime }+x^{2} y^{\prime }+x y = 0 \]
Or
\[ x \left (5 y^{\prime \prime \prime \prime } x^{4}+4 y^{\prime \prime \prime } x^{3}+y^{\prime } x +y\right ) = 0 \]
Or for \(x \neq 0\) the above simplifies to
\[ 5 y^{\prime \prime \prime \prime } x^{4}+4 y^{\prime \prime \prime } x^{3}+y^{\prime } x +y = 0 \]
Solved as higher order Euler type ode
Time used: 0.236 (sec)
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
\[ 5 y^{\prime \prime \prime \prime } x^{4}+4 y^{\prime \prime \prime } x^{3}+y^{\prime } x +y = 0 \]
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 |
| | |
| \(\frac {13}{10}+\frac {\sqrt {6}\, \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}}{60} \pm \frac {\sqrt {6}\, \sqrt {\frac {5 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}\, \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+876 \sqrt {6}\, \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}-308 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}+5420 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}}}}{60} i\) |
\(1\) |
complex conjugate root |
| | |
| \(\frac {13}{10}-\frac {\sqrt {6}\, \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}}{60}+\frac {\sqrt {6}\, \sqrt {\frac {-5 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}\, \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+876 \sqrt {6}\, \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+308 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}-5420 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{{2}/{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{{1}/{3}}}}}}}{60}\) | \(1\) | 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 |
| | |
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}} \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_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}} \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 ) \\
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}+\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_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}-\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*}
Maple step by step solution
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 dsolve solution
Solving time : 0.006 (sec)
Leaf size : 38
dsolve(5*x^5*diff(diff(diff(diff(y(x),x),x),x),x)+4*x^4*diff(diff(diff(y(x),x),x),x)+diff(y(x),x)*x^2+x*y(x) = 0,
y(x),singsol=all)
\[
y = \moverset {4}{\munderset {\textit {\_a} =1}{\sum }}x^{\operatorname {RootOf}\left (5 \textit {\_Z}^{4}-26 \textit {\_Z}^{3}+43 \textit {\_Z}^{2}-21 \textit {\_Z} +1, \operatorname {index} =\textit {\_a} \right )} \textit {\_C}_{\textit {\_a}}
\]
Mathematica DSolve solution
Solving time : 1.419 (sec)
Leaf size : 1931
DSolve[{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],{}},
y[x],x,IncludeSingularSolutions->True]
Too large to display