61.27.13 problem 23

Internal problem ID [12523]
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 : 23
Date solved : Tuesday, January 28, 2025 at 08:02:06 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+2 a x y^{\prime }+\left (b \,x^{4}+a^{2} x^{2}+c x +a \right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.468 (sec). Leaf size: 75

dsolve(diff(y(x),x$2)+2*a*x*diff(y(x),x)+(b*x^4+a^2*x^2+c*x+a)*y(x)=0,y(x), singsol=all)
 

\[ y = {\mathrm e}^{-\frac {x^{2} \left (i \sqrt {b}\, x +\frac {3 a}{2}\right )}{3}} x \left (\operatorname {KummerM}\left (\frac {i c +4 \sqrt {b}}{6 \sqrt {b}}, \frac {4}{3}, \frac {2 i \sqrt {b}\, x^{3}}{3}\right ) c_{1} +\operatorname {KummerU}\left (\frac {i c +4 \sqrt {b}}{6 \sqrt {b}}, \frac {4}{3}, \frac {2 i \sqrt {b}\, x^{3}}{3}\right ) c_{2} \right ) \]

✓ Solution by Mathematica

Time used: 0.191 (sec). Leaf size: 121

DSolve[D[y[x],{x,2}]+2*a*x*D[y[x],x]+(b*x^4+a^2*x^2+c*x+a)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {\sqrt [3]{2} \sqrt [3]{x^3} e^{\frac {1}{6} i x^2 \left (2 \sqrt {b} x+3 i a\right )} \left (c_1 \operatorname {HypergeometricU}\left (\frac {1}{3}-\frac {i c}{6 \sqrt {b}},\frac {2}{3},-\frac {2}{3} i \sqrt {b} x^3\right )+c_2 L_{\frac {i c}{6 \sqrt {b}}-\frac {1}{3}}^{-\frac {1}{3}}\left (-\frac {2}{3} i \sqrt {b} x^3\right )\right )}{x} \]