Internal problem ID [8925]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1346.
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 (a +b \right ) y^{\prime }}{x^{2}}+\frac {\left (x \left (a +b \right )+a b \right ) y}{x^{4}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 25
dsolve(diff(diff(y(x),x),x) = 1/x^2*(a+b)*diff(y(x),x)-((a+b)*x+a*b)/x^4*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} x \,{\mathrm e}^{-\frac {a}{x}}+c_{2} x \,{\mathrm e}^{-\frac {b}{x}} \]
✓ Solution by Mathematica
Time used: 0.064 (sec). Leaf size: 37
DSolve[y''[x] == -(((a*b + (a + b)*x)*y[x])/x^4) + ((a + b)*y'[x])/x^2,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {c_2 x e^{-\frac {a}{x}}}{a-b}+c_1 x e^{-\frac {b}{x}} \\ \end{align*}