Internal problem ID [8703]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1123.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x y^{\prime \prime }-\left (2 a \,x^{2}+1\right ) y^{\prime }+b \,x^{3} y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 45
dsolve(x*diff(diff(y(x),x),x)-(2*a*x^2+1)*diff(y(x),x)+b*x^3*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} {\mathrm e}^{\frac {x^{2} \left (\sqrt {a^{2}-b}+a \right )}{2}}+c_{2} {\mathrm e}^{\frac {x^{2} \left (-\sqrt {a^{2}-b}+a \right )}{2}} \]
✓ Solution by Mathematica
Time used: 0.006 (sec). Leaf size: 53
DSolve[b*x^3*y[x] - (1 + 2*a*x^2)*y'[x] + x*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{\frac {1}{2} x^2 \left (a-\sqrt {a^2-b}\right )} \left (c_2 e^{x^2 \sqrt {a^2-b}}+c_1\right ) \\ \end{align*}