Internal problem ID [11030]
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: 196.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\left (x^{3} a +b \,x^{2}+x c \right ) y^{\prime \prime }+\left (\alpha \,x^{2}+\beta x +2 c \right ) y^{\prime }-\left (\alpha x +2 b -\beta \right ) y=0} \]
✓ Solution by Maple
Time used: 3.5 (sec). Leaf size: 1747
dsolve((a*x^3+b*x^2+c*x)*diff(y(x),x$2)+(alpha*x^2+beta*x+2*c)*diff(y(x),x)-(alpha*x+2*b-beta)*y(x)=0,y(x), singsol=all)
\[ \text {Expression too large to display} \]
✓ Solution by Mathematica
Time used: 6.092 (sec). Leaf size: 224
DSolve[(a*x^3+b*x^2+c*x)*y''[x]+(\[Alpha]*x^2+\[Beta]*x+2*c)*y'[x]-(\[Alpha]*x+2*b-\[Beta])*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {\left (b (2 a x-3 \beta -2 \alpha x)-a \alpha x^2-2 a \beta x+2 b^2+\beta ^2+\alpha c+\alpha ^2 x^2+2 \alpha \beta x\right ) \left (c_2 \int _1^x\frac {\exp \left (\frac {(b \alpha +2 a (b-\beta )) \arctan \left (\frac {b+2 a K[1]}{\sqrt {4 a c-b^2}}\right )}{a \sqrt {4 a c-b^2}}\right ) (c+K[1] (b+a K[1]))^{1-\frac {\alpha }{2 a}}}{\left (2 b^2-3 \beta b+2 (a-\alpha ) K[1] b+\beta ^2+\alpha ^2 K[1]^2-a \alpha K[1]^2+c \alpha -2 a \beta K[1]+2 \alpha \beta K[1]\right )^2}dK[1]+c_1\right )}{x \left (2 b^2+\beta ^2-3 \beta b+\alpha c\right )} \]