Internal problem ID [10876]
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: 42.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+\left (a \,x^{3}+b x \right ) y^{\prime }+2 \left (2 a \,x^{2}+b \right ) y=0} \]
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 84
dsolve(diff(y(x),x$2)+(a*x^3+b*x)*diff(y(x),x)+2*(2*a*x^2+b)*y(x)=0,y(x), singsol=all)
\[ y = c_{1} x \operatorname {HeunB}\left (\frac {1}{2}, \frac {b}{\sqrt {a}}, \frac {5}{2}, -\frac {3 b}{2 \sqrt {a}}, \frac {\sqrt {a}\, x^{2}}{2}\right ) {\mathrm e}^{-\frac {x^{2} \left (a \,x^{2}+2 b \right )}{4}}+c_{2} \operatorname {HeunB}\left (-\frac {1}{2}, \frac {b}{\sqrt {a}}, \frac {5}{2}, -\frac {3 b}{2 \sqrt {a}}, \frac {\sqrt {a}\, x^{2}}{2}\right ) {\mathrm e}^{-\frac {x^{2} \left (a \,x^{2}+2 b \right )}{4}} \]
✓ Solution by Mathematica
Time used: 2.589 (sec). Leaf size: 63
DSolve[y''[x]+(a*x^3+b*x)*y'[x]+2*(2*a*x^2+b)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to x e^{-\frac {1}{4} x^2 \left (a x^2+2 b\right )} \left (c_2 \int _1^x\frac {e^{\frac {1}{4} \left (a K[1]^4+2 b K[1]^2\right )}}{K[1]^2}dK[1]+c_1\right ) \]