2.41 problem 40

Internal problem ID [6424]

Book: Own collection of miscellaneous problems
Section: section 2.0
Problem number: 40.
ODE order: 2.
ODE degree: 1.

CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]

Solve \begin {gather*} \boxed {y^{\prime \prime }-x^{3} y-x^{4}=0} \end {gather*}

Solution by Maple

Time used: 0.032 (sec). Leaf size: 32

dsolve(diff(y(x),x$2)-x^3*y(x)-x^4=0,y(x), singsol=all)
 

\[ y \relax (x ) = \sqrt {x}\, \BesselI \left (\frac {1}{5}, \frac {2 x^{\frac {5}{2}}}{5}\right ) c_{2}+\sqrt {x}\, \BesselK \left (\frac {1}{5}, \frac {2 x^{\frac {5}{2}}}{5}\right ) c_{1}-x \]

Solution by Mathematica

Time used: 0.164 (sec). Leaf size: 109

DSolve[y''[x]-x^3*y[x]-x^4==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {\sqrt {2 \left (1+\sqrt {5}\right )} \pi c_1 \sqrt {x} I_{-\frac {1}{5}}\left (\frac {2 x^{5/2}}{5}\right )}{5^{9/20} \text {Gamma}\left (\frac {1}{5}\right )}+\frac {\text {Gamma}\left (\frac {1}{5}\right ) \left (\left (x^{5/2}\right )^{4/5}+\frac {\sqrt [5]{-1} c_2 x^2}{5^{2/5}}\right ) I_{\frac {1}{5}}\left (\frac {2 x^{5/2}}{5}\right )}{5^{4/5} x^{3/2}}-x \\ \end{align*}