Internal problem ID [8875]
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)]`]]
\[ \boxed {\left (a \,x^{2}+1\right ) y^{\prime \prime }+a x y^{\prime }+y b=0} \]
✓ Solution by Maple
Time used: 0.0 (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)
\[ y \left (x \right ) = c_{1} \left (\sqrt {a}\, x +\sqrt {a \,x^{2}+1}\right )^{\frac {i \sqrt {b}}{\sqrt {a}}}+c_{2} \left (\sqrt {a}\, x +\sqrt {a \,x^{2}+1}\right )^{-\frac {i \sqrt {b}}{\sqrt {a}}} \]
✓ Solution by Mathematica
Time used: 0.074 (sec). Leaf size: 84
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 {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+1}-1}\right )}{\sqrt {a}}\right )+c_2 \sin \left (\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+1}-1}\right )}{\sqrt {a}}\right ) \\ \end{align*}