Internal problem ID [6536]
Book: Own collection of miscellaneous problems
Section: section 4.0
Problem number: 65.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-\left (x^{2}+3\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 36
dsolve(diff(y(x),x$2)=(x^2+3)*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} {\mathrm e}^{\frac {x^{2}}{2}} x +c_{2} \left ({\mathrm e}^{\frac {x^{2}}{2}} \sqrt {\pi }\, \erf \relax (x ) x +{\mathrm e}^{-\frac {x^{2}}{2}}\right ) \]
✓ Solution by Mathematica
Time used: 0.074 (sec). Leaf size: 41
DSolve[y''[x]==(x^2+3)*y[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{-\frac {x^2}{2}} \left (e^{x^2} x \left (c_1-\sqrt {\pi } c_2 \text {Erf}(x)\right )-c_2\right ) \\ \end{align*}