61.31.28 problem 209

Internal problem ID [12709]
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 : 209
Date solved : Tuesday, January 28, 2025 at 08:17:23 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 12.691 (sec). Leaf size: 76

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

\[ y = \left (x +\lambda \right ) \left (\left (\int {\mathrm e}^{-\int \frac {x^{4}+\left (2 a +\lambda \right ) x^{3}+2 b \,x^{2}+\left (\lambda ^{3}+2 c \right ) x +\lambda ^{4}+2 d}{\left (a \,x^{3}+b \,x^{2}+c x +d \right ) \left (x +\lambda \right )}d x}d x \right ) c_{2} +c_{1} \right ) \]

✓ Solution by Mathematica

Time used: 1.700 (sec). Leaf size: 103

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

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