Internal problem ID [11027]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-6 Equation of form
\((a_3 x^3+a_2 x^2 x+a_1 x+a_0) y''+f(x)y'+g(x)y=0\)
Problem number: 193.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} \left (a x +b \right ) y^{\prime \prime }+\left (a \left (2-m -n \right ) x^{2}-b \left (m +n \right ) x \right ) y^{\prime }+\left (a m \left (-1+n \right ) x +b n \left (m +1\right )\right ) y=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 31
dsolve(x^2*(a*x+b)*diff(y(x),x$2)+(a*(2-n-m)*x^2-b*(n+m)*x)*diff(y(x),x)+(a*m*(n-1)*x+b*n*(m+1))*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {c_{1} x^{n}}{a x +b}+\frac {c_{2} x^{m +1}}{a x +b} \]
✓ Solution by Mathematica
Time used: 0.403 (sec). Leaf size: 82
DSolve[x^2*(a*x+b)*y''[x]+(a*(2-n-m)*x^2-b*(n+m)*x)*y'[x]+(a*m*(n-1)*x+b*n*(m+1))*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {x^{\frac {1}{2} \left (-\sqrt {(m-n+1)^2}+m+n+1\right )} \left (c_2 x^{\sqrt {(m-n+1)^2}}+c_1 \sqrt {(m-n+1)^2}\right )}{\sqrt {(m-n+1)^2} (a x+b)} \]