61.31.26 problem 207

Internal problem ID [12707]
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 : 207
Date solved : Tuesday, January 28, 2025 at 08:15:00 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 \left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }+3 \left (3 a \,x^{2}+2 b x +c \right ) y^{\prime }+\left (6 a x +2 b +\lambda \right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 1.701 (sec). Leaf size: 101

dsolve(2*(a*x^3+b*x^2+c*x+d)*diff(y(x),x$2)+3*(3*a*x^2+2*b*x+c)*diff(y(x),x)+(6*a*x+2*b+lambda)*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 96.652 (sec). Leaf size: 3202

DSolve[2*(a*x^3+b*x^2+c*x+d)*D[y[x],{x,2}]+3*(3*a*x^2+2*b*x+c)*D[y[x],x]+(6*a*x+2*b+\[Lambda])*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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