61.31.15 problem 196

Internal problem ID [12696]
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 : 196
Date solved : Tuesday, January 28, 2025 at 08:11:12 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (\alpha \,x^{2}+\beta x +2 c \right ) y^{\prime }-\left (\alpha x +2 b -\beta \right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 2.770 (sec). Leaf size: 1505

dsolve((a*x^3+b*x^2+c*x)*diff(y(x),x$2)+(alpha*x^2+beta*x+2*c)*diff(y(x),x)-(alpha*x+2*b-beta)*y(x)=0,y(x), singsol=all)
 

\[ \text {Expression too large to display} \]

✓ Solution by Mathematica

Time used: 2.829 (sec). Leaf size: 367

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

\[ y(x)\to \frac {\left (b (2 a x-3 \beta -2 \alpha x)-a \alpha x^2-2 a \beta x+2 b^2+\beta ^2+\alpha c+\alpha ^2 x^2+2 \alpha \beta x\right ) \left (c_2 \int _1^x\exp \left (\int _1^{K[2]}\frac {4 b^3-2 (4 \beta +(\alpha -2 a) K[1]) b^2+(6 c \alpha +5 \beta (\beta +\alpha K[1])-4 a (c+2 \beta K[1])) b+2 a^2 \alpha K[1]^3+a \left (-\alpha ^2 K[1]^3+3 \alpha \beta K[1]^2+4 \beta ^2 K[1]+6 c \alpha K[1]+4 c \beta \right )-(\beta +\alpha K[1]) \left ((\beta +\alpha K[1])^2+5 c \alpha \right )}{2 K[1] b^3+(2 c+K[1] (-3 \beta +4 a K[1]-2 \alpha K[1])) b^2+\left (K[1] \left (\beta ^2+(2 \alpha -5 a) K[1] \beta +\left (2 a^2-3 \alpha a+\alpha ^2\right ) K[1]^2\right )-c (3 \beta +(\alpha -2 a) K[1])\right ) b+c^2 \alpha +c \left (\beta ^2+2 (\alpha -a) K[1] \beta +\alpha ^2 K[1]^2\right )+a K[1]^2 \left (\beta ^2+2 (\alpha -a) K[1] \beta +\alpha (\alpha -a) K[1]^2\right )}dK[1]\right )dK[2]+c_1\right )}{x \left (2 b^2+\beta ^2-3 \beta b+\alpha c\right )} \]