61.27.32 problem 42

Internal problem ID [12542]
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
Problem number : 42
Date solved : Tuesday, January 28, 2025 at 03:21:08 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+\left (a \,x^{3}+b x \right ) y^{\prime }+2 \left (2 a \,x^{2}+b \right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 1.576 (sec). Leaf size: 70

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)
 

\[ y = {\mathrm e}^{-\frac {x^{2} \left (a \,x^{2}+2 b \right )}{4}} \left (\operatorname {HeunB}\left (\frac {1}{2}, \frac {b}{\sqrt {a}}, \frac {5}{2}, -\frac {3 b}{2 \sqrt {a}}, \frac {x^{2} \sqrt {a}}{2}\right ) c_{1} x +\operatorname {HeunB}\left (-\frac {1}{2}, \frac {b}{\sqrt {a}}, \frac {5}{2}, -\frac {3 b}{2 \sqrt {a}}, \frac {x^{2} \sqrt {a}}{2}\right ) c_{2} \right ) \]

✓ Solution by Mathematica

Time used: 1.232 (sec). Leaf size: 63

DSolve[D[y[x],{x,2}]+(a*x^3+b*x)*D[y[x],x]+2*(2*a*x^2+b)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to x e^{-\frac {1}{4} x^2 \left (a x^2+2 b\right )} \left (c_2 \int _1^x\frac {e^{\frac {1}{4} \left (a K[1]^4+2 b K[1]^2\right )}}{K[1]^2}dK[1]+c_1\right ) \]