Internal problem ID [10867]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-2 Equation of form
\(y''+f(x)y'+g(x)y=0\)
Problem number: 43.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+\left (a b \,x^{3}+b \,x^{2}+2 a \right ) y^{\prime }+a^{2} \left (x^{3} b +1\right ) y=0} \]
✓ Solution by Maple
Time used: 0.437 (sec). Leaf size: 41
dsolve(diff(y(x),x$2)+(a*b*x^3+b*x^2+2*a)*diff(y(x),x)+a^2*(b*x^3+1)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{-a x} \left (c_{2} \left (\int \frac {{\mathrm e}^{-\frac {b \,x^{3} \left (a x +\frac {4}{3}\right )}{4}}}{\left (a x +1\right )^{2}}d x \right )+c_{1} \right ) \left (a x +1\right ) \]
✓ Solution by Mathematica
Time used: 3.356 (sec). Leaf size: 57
DSolve[y''[x]+(a*b*x^3+b*x^2+2*a)*y'[x]+a^2*(b*x^3+1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-a x} (a x+1) \left (c_2 \int _1^x\frac {e^{-\frac {1}{12} b K[1]^3 (3 a K[1]+4)}}{(a K[1]+1)^2}dK[1]+c_1\right ) \]