Internal problem ID [11023]
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: 199.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (\left (m -a \right ) x^{2}+\left (2 c m -1\right ) x -c \right ) y^{\prime }+\left (-2 m x +1\right ) y=0} \]
✗ Solution by Maple
dsolve((a*x^3+b*x^2+c*x)*diff(y(x),x$2)+((m-a)*x^2+(2*c*m-1)*x-c)*diff(y(x),x)+(-2*m*x+1)*y(x)=0,y(x), singsol=all)
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 17.694 (sec). Leaf size: 192
DSolve[(a*x^3+b*x^2+c*x)*y''[x]+((m-a)*x^2+(2*c*m-1)*x-c)*y'[x]+(-2*m*x+1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {\left (x (a x+2 b+m x-1)+c (2 b+4 m x-1)+4 c^2 m\right ) \left (c_2 \int _1^x\frac {\exp \left (\frac {(b m-2 a (b+2 c m-1)) \arctan \left (\frac {b+2 a K[1]}{\sqrt {4 a c-b^2}}\right )}{a \sqrt {4 a c-b^2}}\right ) K[1] (c+K[1] (b+a K[1]))^{-\frac {m}{2 a}}}{\left (4 m c^2+(2 b+4 m K[1]-1) c+K[1] (2 b+a K[1]+m K[1]-1)\right )^2}dK[1]+c_1\right )}{a+2 b (c+1)+4 c^2 m+4 c m-c+m-1} \]