Internal problem ID [11031]
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 Equation of form
\((a_3 x^3+a_2 x^2 x+a_1 x+a_0) y''+f(x)y'+g(x)y=0\)
Problem number: 207.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {2 \left (a \,x^{3}+b \,x^{2}+c x +d \right ) y^{\prime \prime }+3 \left (3 a \,x^{2}+2 b x +c \right ) y^{\prime }+\left (6 a x +2 b +\lambda \right ) y=0} \]
✓ Solution by Maple
Time used: 0.531 (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)
\[ y \left (x \right ) = \frac {c_{1} {\mathrm e}^{\frac {\sqrt {2}\, \sqrt {-\frac {\lambda }{a}}\, \left (\int \frac {1}{\sqrt {\frac {a \,x^{3}+x^{2} b +c x +d}{a}}}d x \right )}{2}}+c_{2} {\mathrm e}^{-\frac {\sqrt {2}\, \sqrt {-\frac {\lambda }{a}}\, \left (\int \frac {1}{\sqrt {\frac {a \,x^{3}+x^{2} b +c x +d}{a}}}d x \right )}{2}}}{\sqrt {a \,x^{3}+x^{2} b +c x +d}} \]
✓ Solution by Mathematica
Time used: 135.727 (sec). Leaf size: 3202
DSolve[2*(a*x^3+b*x^2+c*x+d)*y''[x]+3*(3*a*x^2+2*b*x+c)*y'[x]+(6*a*x+2*b+\[Lambda])*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
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