Internal problem ID [6419]
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]]
\[ \boxed {y^{\prime \prime }-y x^{2}-x^{4}=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 120
dsolve(diff(y(x),x$2)-x^2*y(x)-x^4=0,y(x), singsol=all)
\[ y \left (x \right ) = \sqrt {x}\, \operatorname {BesselI}\left (\frac {1}{4}, \frac {x^{2}}{2}\right ) c_{2} +\sqrt {x}\, \operatorname {BesselK}\left (\frac {1}{4}, \frac {x^{2}}{2}\right ) c_{1} -\frac {x^{\frac {11}{2}} \left (\pi \Gamma \left (\frac {3}{4}\right )^{2} \operatorname {hypergeom}\left (\left [\frac {3}{2}\right ], \left [\frac {19}{8}, \frac {5}{2}\right ], \frac {x^{4}}{16}\right ) \sqrt {2}\, x \operatorname {BesselI}\left (\frac {1}{4}, \frac {x^{2}}{2}\right )+2 \Gamma \left (\frac {3}{4}\right )^{2} \operatorname {hypergeom}\left (\left [\frac {3}{2}\right ], \left [\frac {5}{4}, \frac {5}{2}\right ], \frac {x^{4}}{16}\right ) x \operatorname {BesselK}\left (\frac {1}{4}, \frac {x^{2}}{2}\right )-\frac {6 \pi ^{2} \operatorname {csgn}\left (x \right ) \operatorname {hypergeom}\left (\left [\frac {5}{4}\right ], \left [\frac {3}{4}, \frac {5}{2}\right ], \frac {x^{4}}{16}\right ) \operatorname {BesselI}\left (\frac {1}{4}, \frac {x^{2}}{2}\right )}{5}\right )}{12 \Gamma \left (\frac {3}{4}\right ) \pi } \]
✓ Solution by Mathematica
Time used: 2.783 (sec). Leaf size: 213
DSolve[y''[x]-x^2*y[x]-x^4==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \operatorname {ParabolicCylinderD}\left (-\frac {1}{2},\sqrt {2} x\right ) \left (\int _1^x\frac {K[1]^4 \operatorname {ParabolicCylinderD}\left (-\frac {1}{2},i \sqrt {2} K[1]\right )}{\sqrt {2} \left (\operatorname {HermiteH}\left (-\frac {1}{2},K[1]\right ) \left (i \operatorname {HermiteH}\left (\frac {1}{2},i K[1]\right )+2 \operatorname {HermiteH}\left (-\frac {1}{2},i K[1]\right ) K[1]\right )-\operatorname {HermiteH}\left (-\frac {1}{2},i K[1]\right ) \operatorname {HermiteH}\left (\frac {1}{2},K[1]\right )\right )}dK[1]+c_1\right )+\operatorname {ParabolicCylinderD}\left (-\frac {1}{2},i \sqrt {2} x\right ) \left (\int _1^x\frac {K[2]^4 \operatorname {ParabolicCylinderD}\left (-\frac {1}{2},\sqrt {2} K[2]\right )}{\sqrt {2} \left (\operatorname {HermiteH}\left (-\frac {1}{2},i K[2]\right ) \operatorname {HermiteH}\left (\frac {1}{2},K[2]\right )+\operatorname {HermiteH}\left (-\frac {1}{2},K[2]\right ) \left (-i \operatorname {HermiteH}\left (\frac {1}{2},i K[2]\right )-2 \operatorname {HermiteH}\left (-\frac {1}{2},i K[2]\right ) K[2]\right )\right )}dK[2]+c_2\right ) \\ \end{align*}