Internal problem ID [10996]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-5 Equation of form
\((a x^2+b x+c) y''+f(x)y'+g(x)y=0\)
Problem number: 162.
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)]`]]
\[ \boxed {\left (a \,x^{2}+b \right ) y^{\prime \prime }+a x y^{\prime }+c y=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 61
dsolve((a*x^2+b)*diff(y(x),x$2)+a*x*diff(y(x),x)+c*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \left (\sqrt {a}\, x +\sqrt {a \,x^{2}+b}\right )^{\frac {i \sqrt {c}}{\sqrt {a}}}+c_{2} \left (\sqrt {a}\, x +\sqrt {a \,x^{2}+b}\right )^{-\frac {i \sqrt {c}}{\sqrt {a}}} \]
✓ Solution by Mathematica
Time used: 0.196 (sec). Leaf size: 74
DSolve[(a*x^2+b)*y''[x]+a*x*y'[x]+c*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 \cos \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b}}\right )}{\sqrt {a}}\right )+c_2 \sin \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b}}\right )}{\sqrt {a}}\right ) \]