Internal problem ID [8877]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1297.
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 {\left (a \,x^{2}+1\right ) y^{\prime \prime }+a x y^{\prime }+b y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.004 (sec). Leaf size: 61
dsolve((a*x^2+1)*diff(diff(y(x),x),x)+a*x*diff(y(x),x)+b*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \left (x \sqrt {a}+\sqrt {a \,x^{2}+1}\right )^{\frac {i \sqrt {b}}{\sqrt {a}}}+c_{2} \left (x \sqrt {a}+\sqrt {a \,x^{2}+1}\right )^{-\frac {i \sqrt {b}}{\sqrt {a}}} \]
✓ Solution by Mathematica
Time used: 0.038 (sec). Leaf size: 52
DSolve[b*y[x] + a*x*y'[x] + (1 + a*x^2)*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 \cos \left (\frac {\sqrt {b} \sinh ^{-1}\left (\sqrt {a} x\right )}{\sqrt {a}}\right )+c_2 \sin \left (\frac {\sqrt {b} \sinh ^{-1}\left (\sqrt {a} x\right )}{\sqrt {a}}\right ) \\ \end{align*}