Internal problem ID [8970]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1391.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-\frac {\left (7 a \,x^{2}+5\right ) y^{\prime }}{x \left (a \,x^{2}+1\right )}+\frac {\left (15 a \,x^{2}+5\right ) y}{x^{2} \left (a \,x^{2}+1\right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 20
dsolve(diff(diff(y(x),x),x) = 1/x*(7*a*x^2+5)/(a*x^2+1)*diff(y(x),x)-(15*a*x^2+5)/x^2/(a*x^2+1)*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} x^{5}+c_{2} \left (2 x^{3} a +x \right ) \]
✓ Solution by Mathematica
Time used: 0.029 (sec). Leaf size: 27
DSolve[y''[x] == -(((5 + 15*a*x^2)*y[x])/(x^2*(1 + a*x^2))) + ((5 + 7*a*x^2)*y'[x])/(x*(1 + a*x^2)),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 x^5-\frac {1}{4} c_2 x \left (2 a x^2+1\right ) \\ \end{align*}