Internal problem ID [10971]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-4 Equation of form
\(x^2 y''+f(x)y'+g(x)y=0\)
Problem number: 137.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }-b y=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 48
dsolve(x^2*diff(y(x),x$2)+(a*x^2+b*x)*diff(y(x),x)-b*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} x^{-b} {\mathrm e}^{-a x}+c_{2} \left (1-b \,{\mathrm e}^{-a x} \left (\Gamma \left (b \right )-\Gamma \left (b , -a x \right )\right ) \left (-a x \right )^{-b}\right ) \]
✓ Solution by Mathematica
Time used: 0.04 (sec). Leaf size: 43
DSolve[x^2*y''[x]+(a*x^2+b*x)*y'[x]-b*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-a x} \left (\frac {c_1 (-a x)^{-b} \Gamma (b+1,-a x)}{a}+c_2 x^{-b}\right ) \]