Internal problem ID [7210]
Internal file name [OUTPUT/6196_Sunday_June_05_2022_04_27_39_PM_75557255/index.tex]
Book: Own collection of miscellaneous problems
Section: section 3.0
Problem number: 20.
ODE order: 4.
ODE degree: 1.
The type(s) of ODE detected by this program : "higher_order_ODE_non_constant_coefficients_of_type_Euler"
Maple gives the following as the ode type
[[_high_order, _with_linear_symmetries]]
\[ \boxed {5 x^{5} y^{\prime \prime \prime \prime }+4 x^{4} y^{\prime \prime \prime }+y^{\prime } x^{2}+y x=0} \] 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 \[ 5 y^{\prime \prime \prime \prime } x^{4}+4 y^{\prime \prime \prime } x^{3}+x y^{\prime }+y = 0 \] 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 )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}}{60}+\frac {i \sqrt {6}\, \sqrt {\frac {5 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}\, \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+876 \sqrt {6}\, \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}-308 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}+5420 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}}}}{60}\\ \lambda _2 &= \frac {13}{10}+\frac {\sqrt {6}\, \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}}{60}-\frac {i \sqrt {6}\, \sqrt {\frac {5 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}\, \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+876 \sqrt {6}\, \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}-308 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}+5420 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}}}}{60}\\ \lambda _3 &= \frac {13}{10}-\frac {\sqrt {6}\, \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}}{60}+\frac {\sqrt {6}\, \sqrt {\frac {-5 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}\, \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+876 \sqrt {6}\, \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+308 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}-5420 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}}}}{60}\\ \lambda _4 &= \frac {13}{10}-\frac {\sqrt {6}\, \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}}{60}-\frac {\sqrt {6}\, \sqrt {\frac {-5 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}\, \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+876 \sqrt {6}\, \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+308 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}-5420 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}}}}{60} \end {align*}
This table summarises the result
| root | multiplicity | type of root |
| \(\frac {13}{10}-\frac {\sqrt {6}\, \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}}{60}+\frac {\sqrt {6}\, \sqrt {\frac {-5 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}\, \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+876 \sqrt {6}\, \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+308 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}-5420 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}}}}{60}\) | \(1\) | real root |
| \(\frac {13}{10}-\frac {\sqrt {6}\, \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}}{60}-\frac {\sqrt {6}\, \sqrt {\frac {-5 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}\, \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+876 \sqrt {6}\, \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+308 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}-5420 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}}}}{60}\) | \(1\) | real root |
| \(\frac {13}{10}+\frac {\sqrt {6}\, \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}}{60} \pm \frac {\sqrt {6}\, \sqrt {\frac {5 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}\, \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+876 \sqrt {6}\, \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}-308 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}+5420 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}}}}{60} i\) | \(1\) | 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 )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}}{60}+\frac {\sqrt {6}\, \sqrt {\frac {-5 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}\, \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+876 \sqrt {6}\, \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+308 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}-5420 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}}}}{60}}\\ y_2 &= x^{\frac {13}{10}-\frac {\sqrt {6}\, \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}}{60}-\frac {\sqrt {6}\, \sqrt {\frac {-5 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}\, \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+876 \sqrt {6}\, \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+308 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}-5420 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}}}}{60}}\\ y_3 &= x^{\frac {13}{10}+\frac {\sqrt {6}\, \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}}{60}} \cos \left (\frac {\sqrt {6}\, \sqrt {\frac {5 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}\, \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+876 \sqrt {6}\, \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}-308 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}+5420 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {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 )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}}{60}} \sin \left (\frac {\sqrt {6}\, \sqrt {\frac {5 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}\, \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+876 \sqrt {6}\, \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}-308 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}+5420 \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}} \sqrt {\frac {5 \left (40076+12 \sqrt {2307813}\right )^{\frac {2}{3}}+154 \left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}+5420}{\left (40076+12 \sqrt {2307813}\right )^{\frac {1}{3}}}}}}\, \ln \left (x \right )}{60}\right ) \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \text {Expression too large to display} \\ \end{align*}
Verification of solutions
\[ \text {Expression too large to display} \] Verified OK.
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`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 38
dsolve(5*x^5*diff(y(x),x$4)+4*x^4*diff(y(x),x$3)+x^2*diff(y(x),x)+x*y(x)= 0,y(x), singsol=all)
\[ y \left (x \right ) = \moverset {4}{\munderset {\textit {\_a} =1}{\sum }}x^{\operatorname {RootOf}\left (5 \textit {\_Z}^{4}-26 \textit {\_Z}^{3}+43 \textit {\_Z}^{2}-21 \textit {\_Z} +1, \operatorname {index} =\textit {\_a} \right )} \textit {\_C}_{\textit {\_a}} \]
✓ Solution by Mathematica
Time used: 1.114 (sec). Leaf size: 1931
DSolve[5*x^5*y''''[x]+4*x^4*y'''[x]+x^2*y'[x]+x*y[x]== Sin[x],y[x],x,IncludeSingularSolutions -> True]
Too large to display