Internal problem ID [6419]
Book: Own collection of miscellaneous problems
Section: section 2.0
Problem number: 35.
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^{2} y-x^{3}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 32
dsolve(diff(y(x),x$2)-x^2*y(x)-x^3=0,y(x), singsol=all)
\[ y \relax (x ) = \sqrt {x}\, \BesselI \left (\frac {1}{4}, \frac {x^{2}}{2}\right ) c_{2}+\sqrt {x}\, \BesselK \left (\frac {1}{4}, \frac {x^{2}}{2}\right ) c_{1}-x \]
✓ Solution by Mathematica
Time used: 4.308 (sec). Leaf size: 213
DSolve[y''[x]-x^2*y[x]-x^3==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]^3 D_{-\frac {1}{2}}\left (i \sqrt {2} K[1]\right )}{\sqrt {2} \left (\text {HermiteH}\left (-\frac {1}{2},K[1]\right ) \left (i \text {HermiteH}\left (\frac {1}{2},i K[1]\right )+2 \text {HermiteH}\left (-\frac {1}{2},i K[1]\right ) K[1]\right )-\text {HermiteH}\left (-\frac {1}{2},i K[1]\right ) \text {HermiteH}\left (\frac {1}{2},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]^3 D_{-\frac {1}{2}}\left (\sqrt {2} K[2]\right )}{\sqrt {2} \left (\text {HermiteH}\left (-\frac {1}{2},i K[2]\right ) \text {HermiteH}\left (\frac {1}{2},K[2]\right )+\text {HermiteH}\left (-\frac {1}{2},K[2]\right ) \left (-i \text {HermiteH}\left (\frac {1}{2},i K[2]\right )-2 \text {HermiteH}\left (-\frac {1}{2},i K[2]\right ) K[2]\right )\right )}dK[2]+c_2\right ) \\ \end{align*}