61.28.29 problem 89

Internal problem ID [12589]
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 : 89
Date solved : Tuesday, January 28, 2025 at 03:22:26 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 12.417 (sec). Leaf size: 48

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

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

✓ Solution by Mathematica

Time used: 3.123 (sec). Leaf size: 78

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

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