Internal problem ID [10857]
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: 23.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+2 a x y^{\prime }+\left (b \,x^{4}+a^{2} x^{2}+x c +a \right ) y=0} \]
✓ Solution by Maple
Time used: 0.25 (sec). Leaf size: 97
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)
\[ y \left (x \right ) = c_{1} x \,{\mathrm e}^{-\frac {x^{2} \left (i \sqrt {b}\, x +\frac {3 a}{2}\right )}{3}} \operatorname {KummerM}\left (\frac {i c +4 \sqrt {b}}{6 \sqrt {b}}, \frac {4}{3}, \frac {2 i \sqrt {b}\, x^{3}}{3}\right )+c_{2} x \,{\mathrm e}^{-\frac {x^{2} \left (i \sqrt {b}\, x +\frac {3 a}{2}\right )}{3}} \operatorname {KummerU}\left (\frac {i c +4 \sqrt {b}}{6 \sqrt {b}}, \frac {4}{3}, \frac {2 i \sqrt {b}\, x^{3}}{3}\right ) \]
✓ Solution by Mathematica
Time used: 0.322 (sec). Leaf size: 121
DSolve[y''[x]+2*a*x*y'[x]+(b*x^4+a^2*x^2+c*x+a)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {\sqrt [3]{2} \sqrt [3]{x^3} e^{\frac {1}{6} i x^2 \left (2 \sqrt {b} x+3 i a\right )} \left (c_1 \operatorname {HypergeometricU}\left (\frac {1}{3}-\frac {i c}{6 \sqrt {b}},\frac {2}{3},-\frac {2}{3} i \sqrt {b} x^3\right )+c_2 L_{\frac {i c}{6 \sqrt {b}}-\frac {1}{3}}^{-\frac {1}{3}}\left (-\frac {2}{3} i \sqrt {b} x^3\right )\right )}{x} \]