Internal problem ID [6420]
Book: Own collection of miscellaneous problems
Section: section 2.0
Problem number: 36.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-y x^{2}-x^{4}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.012 (sec). Leaf size: 120
dsolve(diff(y(x),x$2)-x^2*y(x)-x^4=0,y(x), singsol=all)
\[ y \relax (x ) = \sqrt {x}\, \BesselI \left (\frac {1}{4}, \frac {x^{2}}{2}\right ) c_{2}+\sqrt {x}\, \BesselK \left (\frac {1}{4}, \frac {x^{2}}{2}\right ) c_{1}-\frac {\left (\sqrt {2}\, \pi x \BesselI \left (\frac {1}{4}, \frac {x^{2}}{2}\right ) \Gamma \left (\frac {3}{4}\right )^{2} \hypergeom \left (\left [\frac {3}{2}\right ], \left [\frac {19}{8}, \frac {5}{2}\right ], \frac {x^{4}}{16}\right )+2 x \BesselK \left (\frac {1}{4}, \frac {x^{2}}{2}\right ) \Gamma \left (\frac {3}{4}\right )^{2} \hypergeom \left (\left [\frac {3}{2}\right ], \left [\frac {5}{4}, \frac {5}{2}\right ], \frac {x^{4}}{16}\right )-\frac {6 \pi ^{2} \BesselI \left (\frac {1}{4}, \frac {x^{2}}{2}\right ) \mathrm {csgn}\relax (x ) \hypergeom \left (\left [\frac {5}{4}\right ], \left [\frac {3}{4}, \frac {5}{2}\right ], \frac {x^{4}}{16}\right )}{5}\right ) x^{\frac {11}{2}}}{12 \Gamma \left (\frac {3}{4}\right ) \pi } \]
✓ Solution by Mathematica
Time used: 2.417 (sec). Leaf size: 203
DSolve[y''[x]-x^2*y[x]-x^4==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to D_{-\frac {1}{2}}\left (\sqrt {2} x\right ) \left (\int _1^x-\frac {K[1]^4 D_{-\frac {1}{2}}\left (i \sqrt {2} K[1]\right )}{\sqrt {2} \left (\text {HermiteH}\left (-\frac {1}{2},i K[1]\right ) \left (\text {HermiteH}\left (\frac {1}{2},K[1]\right )-2 \text {HermiteH}\left (-\frac {1}{2},K[1]\right ) K[1]\right )-i \text {HermiteH}\left (-\frac {1}{2},K[1]\right ) \text {HermiteH}\left (\frac {1}{2},i K[1]\right )\right )}dK[1]+c_1\right )+D_{-\frac {1}{2}}\left (i \sqrt {2} x\right ) \left (\int _1^x\frac {K[2]^4 D_{-\frac {1}{2}}\left (\sqrt {2} K[2]\right )}{\sqrt {2} \left (\text {HermiteH}\left (-\frac {1}{2},i K[2]\right ) \left (\text {HermiteH}\left (\frac {1}{2},K[2]\right )-2 \text {HermiteH}\left (-\frac {1}{2},K[2]\right ) K[2]\right )-i \text {HermiteH}\left (-\frac {1}{2},K[2]\right ) \text {HermiteH}\left (\frac {1}{2},i K[2]\right )\right )}dK[2]+c_2\right ) \\ \end{align*}