Internal problem ID [11023]
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: 189.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _exact, _linear, _homogeneous]]
\[ \boxed {x \left (a \,x^{2}+b \right ) y^{\prime \prime }+2 \left (a \,x^{2}+b \right ) y^{\prime }-2 y a x=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 49
dsolve(x*(a*x^2+b)*diff(y(x),x$2)+2*(a*x^2+b)*diff(y(x),x)-2*a*x*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {c_{1} \left (a \,x^{2}+b \right )}{x}+\frac {c_{2} \left (\left (a \,x^{2}+b \right ) \arctan \left (\frac {\sqrt {a b}\, x}{b}\right )+\sqrt {a b}\, x \right )}{x} \]
✓ Solution by Mathematica
Time used: 0.16 (sec). Leaf size: 78
DSolve[x*(a*x^2+b)*y''[x]+2*(a*x^2+b)*y'[x]-2*a*x*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {c_2 \left (a x^2+b\right ) \arctan \left (\frac {\sqrt {a} x}{\sqrt {b}}\right )+\sqrt {a} \sqrt {b} \left (2 a b c_1 x^2+2 b^2 c_1+c_2 x\right )}{2 \sqrt {a} b^{3/2} x} \]