Internal problem ID [6422]
Book: Own collection of miscellaneous problems
Section: section 2.0
Problem number: 39.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }-y x^{3}-x^{3}=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 30
dsolve(diff(y(x),x$2)-x^3*y(x)-x^3=0,y(x), singsol=all)
\[ y \left (x \right ) = \sqrt {x}\, \operatorname {BesselI}\left (\frac {1}{5}, \frac {2 x^{\frac {5}{2}}}{5}\right ) c_{2} +\sqrt {x}\, \operatorname {BesselK}\left (\frac {1}{5}, \frac {2 x^{\frac {5}{2}}}{5}\right ) c_{1} -1 \]
✓ Solution by Mathematica
Time used: 0.124 (sec). Leaf size: 217
DSolve[y''[x]-x^3*y[x]-x^3==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [5]{-1} \operatorname {Gamma}\left (\frac {4}{5}\right ) \left (5^{4/5} x^5 \operatorname {Gamma}\left (\frac {6}{5}\right ) \, _0\tilde {F}_1\left (;\frac {9}{5};\frac {x^5}{25}\right ) \operatorname {BesselI}\left (\frac {1}{5},\frac {2 x^{5/2}}{5}\right )+5\ 5^{2/5} \left (x^{5/2}\right )^{2/5} \operatorname {BesselI}\left (-\frac {1}{5},\frac {2 x^{5/2}}{5}\right )-5^{3/5} \left (x^{5/2}\right )^{6/5} \operatorname {Gamma}\left (\frac {1}{5}\right ) \operatorname {BesselI}\left (-\frac {4}{5},\frac {2 x^{5/2}}{5}\right ) \operatorname {BesselI}\left (-\frac {1}{5},\frac {2 x^{5/2}}{5}\right )\right )}{25 \sqrt [5]{x^{5/2}} \text {Root}\left [25 \text {$\#$1}^5+1\&,5\right ]}+\frac {c_1 \sqrt {x} \operatorname {Gamma}\left (\frac {4}{5}\right ) \operatorname {BesselI}\left (-\frac {1}{5},\frac {2 x^{5/2}}{5}\right )}{\sqrt [5]{5}}+\sqrt [5]{-\frac {1}{5}} c_2 \sqrt {x} \operatorname {Gamma}\left (\frac {6}{5}\right ) \operatorname {BesselI}\left (\frac {1}{5},\frac {2 x^{5/2}}{5}\right ) \\ \end{align*}