61.31.18 problem 199

Internal problem ID [12699]
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
Problem number : 199
Date solved : Tuesday, January 28, 2025 at 03:39:36 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \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 \end{align*}

✗ 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: 11.161 (sec). Leaf size: 277

DSolve[(a*x^3+b*x^2+c*x)*D[y[x],{x,2}]+((m-a)*x^2+(2*c*m-1)*x-c)*D[y[x],x]+(-2*m*x+1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {\left (x (a x+2 b+m x-1)+c (2 b+4 m x-1)+4 c^2 m\right ) \left (c_2 \int _1^x\exp \left (\int _1^{K[2]}\frac {4 m (1-2 m K[1]) c^3+\left (-12 m^2 K[1]^2+2 m (2 a K[1]+1) K[1]+b (2-4 m K[1])-1\right ) c^2-2 K[1]^2 (a (-b+3 m K[1]+2)+m (7 b+3 m K[1]-2)) c-K[1]^2 \left (4 b^2+(6 a K[1]+6 m K[1]-4) b+3 a^2 K[1]^2+(m K[1]-1)^2+2 a K[1] (2 m K[1]-1)\right )}{K[1] (c+K[1] (b+a K[1])) \left (4 m c^2+(2 b+4 m K[1]-1) c+K[1] (2 b+a K[1]+m K[1]-1)\right )}dK[1]\right )dK[2]+c_1\right )}{a+2 b (c+1)+4 c^2 m+4 c m-c+m-1} \]