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: 197.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\left (x^{3} a +b \,x^{2}+x c \right ) y^{\prime \prime }+\left (-2 a \,x^{2}-\left (1+b \right ) x +k \right ) y^{\prime }+2 \left (a x +1\right ) y=0} \]
✓ Solution by Maple
Time used: 0.297 (sec). Leaf size: 2565
dsolve((a*x^3+b*x^2+c*x)*diff(y(x),x$2)+(-2*a*x^2-(b+1)*x+k)*diff(y(x),x)+2*(a*x+1)*y(x)=0,y(x), singsol=all)
\[ \text {Expression too large to display} \]
✓ Solution by Mathematica
Time used: 10.126 (sec). Leaf size: 186
DSolve[(a*x^3+b*x^2+c*x)*y''[x]+(-2*a*x^2-(b+1)*x+k)*y'[x]+2*(a*x+1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -\frac {\left (-k x (a x+2)-(b-1) x^2+c (k-2 x)+k^2\right ) \left (c_2 \int _1^x\frac {\exp \left (\frac {(2 c+b k) \arctan \left (\frac {b+2 a K[1]}{\sqrt {4 a c-b^2}}\right )}{c \sqrt {4 a c-b^2}}\right ) K[1]^{-\frac {k}{c}} (c+K[1] (b+a K[1]))^{\frac {k}{2 c}+1}}{\left (k^2-K[1] (a K[1]+2) k-(b-1) K[1]^2+c (k-2 K[1])\right )^2}dK[1]+c_1\right )}{a k+b-c (k-2)-k^2+2 k-1} \]