Internal problem ID [5820]
Internal file name [OUTPUT/5068_Sunday_June_05_2022_03_19_59_PM_84530589/index.tex]
Book: Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold
Scientific. Singapore. 1995
Section: Chapter 2. Linear homogeneous equations. Section 2.2 problems. page 95
Problem number: 59.
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]]
\[ \boxed {x^{4} y^{\prime \prime \prime \prime }-x^{2} y^{\prime \prime }+y=0} \] 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 \[ x^{4} y^{\prime \prime \prime \prime }-x^{2} y^{\prime \prime }+y = 0 \] gives \[ -x^{2} \lambda \left (\lambda -1\right ) x^{\lambda -2}+x^{4} \lambda \left (\lambda -1\right ) \left (\lambda -2\right ) \left (\lambda -3\right ) x^{\lambda -4}+x^{\lambda } = 0 \] Which simplifies to \[ -\lambda \left (\lambda -1\right ) x^{\lambda }+\lambda \left (\lambda -1\right ) \left (\lambda -2\right ) \left (\lambda -3\right ) x^{\lambda }+x^{\lambda } = 0 \] And since \(x^{\lambda }\neq 0\) then dividing through by \(x^{\lambda }\), the above becomes
\[ -\lambda \left (\lambda -1\right )+\lambda \left (\lambda -1\right ) \left (\lambda -2\right ) \left (\lambda -3\right )+1 = 0 \] Simplifying gives the characteristic equation as \[ \lambda ^{4}-6 \lambda ^{3}+10 \lambda ^{2}-5 \lambda +1 = 0 \] Solving the above gives the following roots \begin {align*} \lambda _1 &= \frac {3}{2}+\frac {\sqrt {6}\, \sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}}{12}+\frac {\sqrt {6}\, \sqrt {\frac {-\sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}\, \left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+24 \sqrt {6}\, \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+28 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}} \sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}-88 \sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}} \sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}}}}{12}\\ \lambda _2 &= \frac {3}{2}+\frac {\sqrt {6}\, \sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}}{12}-\frac {\sqrt {6}\, \sqrt {\frac {-\sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}\, \left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+24 \sqrt {6}\, \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+28 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}} \sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}-88 \sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}} \sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}}}}{12}\\ \lambda _3 &= \frac {3}{2}-\frac {\sqrt {6}\, \sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}}{12}+\frac {i \sqrt {6}\, \sqrt {\frac {\sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}\, \left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+24 \sqrt {6}\, \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}-28 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}} \sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}+88 \sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}} \sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}}}}{12}\\ \lambda _4 &= \frac {3}{2}-\frac {\sqrt {6}\, \sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}}{12}-\frac {i \sqrt {6}\, \sqrt {\frac {\sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}\, \left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+24 \sqrt {6}\, \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}-28 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}} \sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}+88 \sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}} \sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}}}}{12} \end {align*}
This table summarises the result
| root | multiplicity | type of root |
| \(\frac {3}{2}-\frac {\sqrt {6}\, \sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}}{12} \pm \frac {\sqrt {6}\, \sqrt {\frac {\sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}\, \left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+24 \sqrt {6}\, \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}-28 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}} \sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}+88 \sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}} \sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}}}}{12} i\) | \(1\) | complex conjugate root |
| \(\frac {3}{2}+\frac {\sqrt {6}\, \sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}}{12}-\frac {\sqrt {6}\, \sqrt {\frac {-\sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}\, \left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+24 \sqrt {6}\, \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+28 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}} \sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}-88 \sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}} \sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}}}}{12}\) | \(1\) | real root |
| \(\frac {3}{2}+\frac {\sqrt {6}\, \sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}}{12}+\frac {\sqrt {6}\, \sqrt {\frac {-\sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}\, \left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+24 \sqrt {6}\, \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+28 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}} \sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}-88 \sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}} \sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}}}}{12}\) | \(1\) | 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 {3}{2}-\frac {\sqrt {6}\, \sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}}{12}} \cos \left (\frac {\sqrt {6}\, \sqrt {\frac {\sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}\, \left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+24 \sqrt {6}\, \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}-28 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}} \sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}+88 \sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}} \sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}}}\, \ln \left (x \right )}{12}\right )\\ y_2 &= x^{\frac {3}{2}-\frac {\sqrt {6}\, \sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}}{12}} \sin \left (\frac {\sqrt {6}\, \sqrt {\frac {\sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}\, \left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+24 \sqrt {6}\, \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}-28 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}} \sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}+88 \sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}} \sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}}}\, \ln \left (x \right )}{12}\right )\\ y_3 &= x^{\frac {3}{2}+\frac {\sqrt {6}\, \sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}}{12}-\frac {\sqrt {6}\, \sqrt {\frac {-\sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}\, \left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+24 \sqrt {6}\, \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+28 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}} \sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}-88 \sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}} \sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}}}}{12}}\\ y_4 &= x^{\frac {3}{2}+\frac {\sqrt {6}\, \sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}}{12}+\frac {\sqrt {6}\, \sqrt {\frac {-\sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}\, \left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+24 \sqrt {6}\, \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+28 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}} \sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}-88 \sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}} \sqrt {\frac {\left (908+12 \sqrt {993}\right )^{\frac {2}{3}}+14 \left (908+12 \sqrt {993}\right )^{\frac {1}{3}}+88}{\left (908+12 \sqrt {993}\right )^{\frac {1}{3}}}}}}}{12}} \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: 36
dsolve(x^4*diff(y(x),x$4)-x^2*diff(y(x),x$2)+y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \moverset {4}{\munderset {\textit {\_a} =1}{\sum }}x^{\operatorname {RootOf}\left (\textit {\_Z}^{4}-6 \textit {\_Z}^{3}+10 \textit {\_Z}^{2}-5 \textit {\_Z} +1, \operatorname {index} =\textit {\_a} \right )} \textit {\_C}_{\textit {\_a}} \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 130
DSolve[x^4*y''''[x]-x^2*y''[x]+y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_4 x^{\text {Root}\left [\text {$\#$1}^4-6 \text {$\#$1}^3+10 \text {$\#$1}^2-5 \text {$\#$1}+1\&,4\right ]}+c_3 x^{\text {Root}\left [\text {$\#$1}^4-6 \text {$\#$1}^3+10 \text {$\#$1}^2-5 \text {$\#$1}+1\&,3\right ]}+c_1 x^{\text {Root}\left [\text {$\#$1}^4-6 \text {$\#$1}^3+10 \text {$\#$1}^2-5 \text {$\#$1}+1\&,1\right ]}+c_2 x^{\text {Root}\left [\text {$\#$1}^4-6 \text {$\#$1}^3+10 \text {$\#$1}^2-5 \text {$\#$1}+1\&,2\right ]} \]