Internal problem ID [8916]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1337.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, _with_symmetry_[0,F(x)]]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\frac {\left (3 x +a +2 b \right ) y^{\prime }}{2 \left (x +a \right ) \left (x +b \right )}+\frac {\left (a -b \right ) y}{4 \left (x +a \right )^{2} \left (x +b \right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 44
dsolve(diff(diff(y(x),x),x) = -1/2/(x+a)*(3*x+a+2*b)/(x+b)*diff(y(x),x)-1/4*(a-b)/(x+a)^2/(x+b)*y(x),y(x), singsol=all)
\[ y \relax (x ) = \frac {\sqrt {x +b}\, c_{1}}{\sqrt {1+\frac {x +b}{a -b}}}+\frac {c_{2}}{\sqrt {1+\frac {x +b}{a -b}}} \]
✓ Solution by Mathematica
Time used: 0.053 (sec). Leaf size: 53
DSolve[y''[x] == -1/4*((a - b)*y[x])/((a + x)^2*(b + x)) - ((a + 2*b + 3*x)*y'[x])/(2*(a + x)*(b + x)),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {c_1 \sqrt {a-b}+c_2 \sqrt {b+x}}{\sqrt {a-b} \sqrt {\frac {a+x}{a-b}}} \\ \end{align*}