Internal problem ID [6422]
Book: Own collection of miscellaneous problems
Section: section 2.0
Problem number: 38.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-2 y x^{2}-x^{4}+1=0} \end {gather*}
✓ Solution by Maple
Time used: 0.009 (sec). Leaf size: 40
dsolve(diff(y(x),x$2)-2*x^2*y(x)-x^4+1=0,y(x), singsol=all)
\[ y \relax (x ) = \sqrt {x}\, \BesselI \left (\frac {1}{4}, \frac {\sqrt {2}\, x^{2}}{2}\right ) c_{2}+\sqrt {x}\, \BesselK \left (\frac {1}{4}, \frac {\sqrt {2}\, x^{2}}{2}\right ) c_{1}-\frac {x^{2}}{2} \]
✓ Solution by Mathematica
Time used: 1.815 (sec). Leaf size: 288
DSolve[y''[x]-2*x^2*y[x]-x^4+1==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to D_{-\frac {1}{2}}\left (2^{3/4} x\right ) \left (\int _1^x\frac {\left (K[1]^4-1\right ) D_{-\frac {1}{2}}\left (i 2^{3/4} K[1]\right )}{i 2^{3/4} \text {HermiteH}\left (-\frac {1}{2},\sqrt [4]{2} K[1]\right ) \text {HermiteH}\left (\frac {1}{2},i \sqrt [4]{2} K[1]\right )+D_{-\frac {1}{2}}\left (i 2^{3/4} K[1]\right ) \left (2 \sqrt {2} K[1] D_{-\frac {1}{2}}\left (2^{3/4} K[1]\right )-2^{3/4} D_{\frac {1}{2}}\left (2^{3/4} K[1]\right )\right )}dK[1]+c_1\right )+D_{-\frac {1}{2}}\left (i 2^{3/4} x\right ) \left (\int _1^x\frac {i \left (K[2]^4-1\right ) D_{-\frac {1}{2}}\left (2^{3/4} K[2]\right )}{2^{3/4} \text {HermiteH}\left (-\frac {1}{2},\sqrt [4]{2} K[2]\right ) \text {HermiteH}\left (\frac {1}{2},i \sqrt [4]{2} K[2]\right )+i D_{-\frac {1}{2}}\left (i 2^{3/4} K[2]\right ) \left (2^{3/4} D_{\frac {1}{2}}\left (2^{3/4} K[2]\right )-2 \sqrt {2} K[2] D_{-\frac {1}{2}}\left (2^{3/4} K[2]\right )\right )}dK[2]+c_2\right ) \\ \end{align*}