Internal problem ID [8973]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1394.
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 {2 y^{\prime }}{x}+\frac {c y}{x^{2} \left (a x +b \right )^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.008 (sec). Leaf size: 89
dsolve(diff(diff(y(x),x),x) = -2/x*diff(y(x),x)-c/x^2/(a*x+b)^2*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} \sqrt {\frac {a x +b}{x}}\, \left (\frac {x}{a x +b}\right )^{\frac {\sqrt {\frac {b^{2}-4 c}{a^{2}}}\, a}{2 b}}+c_{2} \sqrt {\frac {a x +b}{x}}\, \left (\frac {x}{a x +b}\right )^{-\frac {\sqrt {\frac {b^{2}-4 c}{a^{2}}}\, a}{2 b}} \]
✓ Solution by Mathematica
Time used: 0.03 (sec). Leaf size: 73
DSolve[y''[x] == -((c*y[x])/(x^2*(b + a*x)^2)) - (2*y'[x])/x,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \left (c_2 e^{\frac {\sqrt {b^2-4 c} (\log (x)-\log (a x+b))}{b}}+c_1\right ) \exp \left (-\frac {\left (\sqrt {b^2-4 c}+b\right ) (\log (x)-\log (a x+b))}{2 b}\right ) \\ \end{align*}