Internal problem ID [10090]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-1 Equation of form
\(y''+f(x)y'+g(x)y=0\)
Problem number: 13.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+a y^{\prime }-\left (b \,x^{2}+c \right ) y=0} \]
✓ Solution by Maple
Time used: 0.187 (sec). Leaf size: 85
dsolve(diff(y(x),x$2)+a*diff(y(x),x)-(b*x^2+c)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} x \operatorname {KummerM}\left (\frac {a^{2}+12 \sqrt {b}+4 c}{16 \sqrt {b}}, \frac {3}{2}, \sqrt {b}\, x^{2}\right ) {\mathrm e}^{-\frac {x \left (\sqrt {b}\, x +a \right )}{2}}+c_{2} x \operatorname {KummerU}\left (\frac {a^{2}+12 \sqrt {b}+4 c}{16 \sqrt {b}}, \frac {3}{2}, \sqrt {b}\, x^{2}\right ) {\mathrm e}^{-\frac {x \left (\sqrt {b}\, x +a \right )}{2}} \]
✓ Solution by Mathematica
Time used: 0.011 (sec). Leaf size: 94
DSolve[y''[x]+a*y'[x]-(b*x^2+c)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{-\frac {1}{2} x \left (a+\sqrt {b} x\right )} \left (c_1 \operatorname {HermiteH}\left (-\frac {a^2+4 \left (c+\sqrt {b}\right )}{8 \sqrt {b}},\sqrt [4]{b} x\right )+c_2 \operatorname {Hypergeometric1F1}\left (\frac {a^2+4 \left (c+\sqrt {b}\right )}{16 \sqrt {b}},\frac {1}{2},\sqrt {b} x^2\right )\right ) \\ \end{align*}