Internal problem ID [11021]
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: 187.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{3} y^{\prime \prime }+\left (x^{3} a +a b x -x^{2}+b \right ) y^{\prime }+a^{2} b x y=0} \]
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 59
dsolve(x^3*diff(y(x),x$2)+(a*x^3-x^2+a*b*x+b)*diff(y(x),x)+a^2*b*x*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{-a x} \left (a x +1\right )+c_{2} {\mathrm e}^{-a x} \left (a x +1\right ) \left (\int \frac {x \,{\mathrm e}^{\frac {2 a \,x^{3}+2 a b x +b}{2 x^{2}}}}{\left (a x +1\right )^{2}}d x \right ) \]
✓ Solution by Mathematica
Time used: 1.347 (sec). Leaf size: 70
DSolve[x^3*y''[x]+(a*x^3-x^2+a*b*x+b)*y'[x]+a^2*b*x*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {e^{-a x} (a x+1) \left (c_2 \int _1^x\frac {a^2 e^{a K[1]+\frac {2 a K[1] b+b}{2 K[1]^2}} K[1]}{(a K[1]+1)^2}dK[1]+c_1\right )}{a} \]