Internal problem ID [8955]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1376.
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 (2 x^{2}+a \right ) y^{\prime }}{x \left (x^{2}+a \right )}+\frac {b y}{x^{2} \left (x^{2}+a \right )}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 71
dsolve(diff(diff(y(x),x),x) = -1/x*(2*x^2+a)/(x^2+a)*diff(y(x),x)-b/x^2/(x^2+a)*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} \left (\frac {2 a +2 \sqrt {a}\, \sqrt {x^{2}+a}}{x}\right )^{-\frac {i \sqrt {b}}{\sqrt {a}}}+c_{2} \left (\frac {2 a +2 \sqrt {a}\, \sqrt {x^{2}+a}}{x}\right )^{\frac {i \sqrt {b}}{\sqrt {a}}} \]
✓ Solution by Mathematica
Time used: 0.035 (sec). Leaf size: 69
DSolve[y''[x] == -((b*y[x])/(x^2*(a + x^2))) - ((a + 2*x^2)*y'[x])/(x*(a + x^2)),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 \cos \left (\frac {\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {a+x^2}}{\sqrt {a}}\right )}{\sqrt {a}}\right )-c_2 \sin \left (\frac {\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {a+x^2}}{\sqrt {a}}\right )}{\sqrt {a}}\right ) \\ \end{align*}