Internal problem ID [11025]
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: 191.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} \left (a x +b \right ) y^{\prime \prime }+\left (c \,x^{2}+\left (\lambda a +2 b \right ) x +b \lambda \right ) y^{\prime }+\lambda \left (-2 a +c \right ) y=0} \]
✓ Solution by Maple
Time used: 0.109 (sec). Leaf size: 203
dsolve(x^2*(a*x+b)*diff(y(x),x$2)+(c*x^2+(2*b+a*lambda)*x+b*lambda)*diff(y(x),x)+lambda*(c-2*a)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \left (a x +b \right )^{\frac {3 a -c}{a}} x^{\frac {-3 a +c}{a}} \operatorname {HeunC}\left (\frac {\lambda a}{b}, \frac {a -c}{a}, \frac {3 a -c}{a}, 0, \frac {-2 a^{3} \lambda +\left (c \lambda +5 b \right ) a^{2}-4 c b a +c^{2} b}{2 a^{2} b}, -\frac {b}{a x}\right )+\frac {c_{2} \left (a x +b \right )^{\frac {3 a -c}{a}} \operatorname {HeunC}\left (\frac {\lambda a}{b}, \frac {-a +c}{a}, \frac {3 a -c}{a}, 0, \frac {-2 a^{3} \lambda +\left (c \lambda +5 b \right ) a^{2}-4 c b a +c^{2} b}{2 a^{2} b}, -\frac {b}{a x}\right )}{x^{2}} \]
✓ Solution by Mathematica
Time used: 1.48 (sec). Leaf size: 55
DSolve[x^2*(a*x+b)*y''[x]+(c*x^2+(2*b+a*\[Lambda])*x+b*\[Lambda])*y'[x]+\[Lambda]*(c-2*a)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{\frac {\lambda }{x}} \left (c_2 \int _1^x\frac {e^{-\frac {\lambda }{K[1]}} (b+a K[1])^{2-\frac {c}{a}}}{K[1]^2}dK[1]+c_1\right ) \]