61.28.30 problem 90

Internal problem ID [12590]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-3
Problem number : 90
Date solved : Tuesday, January 28, 2025 at 03:22:28 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }+\left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime }+\left (d -1\right ) \left (a \,x^{2}+b x +c \right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 1.388 (sec). Leaf size: 42

dsolve(x*diff(y(x),x$2)+(a*x^3+b*x^2+c*x+d)*diff(y(x),x)+(d-1)*(a*x^2+b*x+c)*y(x)=0,y(x), singsol=all)
 

\[ y = x^{-d +1} \left (\left (\int x^{d -2} {\mathrm e}^{-\frac {1}{3} a \,x^{3}-\frac {1}{2} b \,x^{2}-c x}d x \right ) c_{2} +c_{1} \right ) \]

✓ Solution by Mathematica

Time used: 0.638 (sec). Leaf size: 63

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

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