61.27.3 problem 13

Internal problem ID [12513]
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 : 13
Date solved : Tuesday, January 28, 2025 at 08:01:59 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.246 (sec). Leaf size: 74

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

\[ y = {\mathrm e}^{-\frac {x \left (\sqrt {b}\, x +a \right )}{2}} x \left (\operatorname {KummerM}\left (\frac {a^{2}+12 \sqrt {b}+4 c}{16 \sqrt {b}}, \frac {3}{2}, \sqrt {b}\, x^{2}\right ) c_{1} +\operatorname {KummerU}\left (\frac {a^{2}+12 \sqrt {b}+4 c}{16 \sqrt {b}}, \frac {3}{2}, \sqrt {b}\, x^{2}\right ) c_{2} \right ) \]

✓ Solution by Mathematica

Time used: 0.051 (sec). Leaf size: 96

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

\[ y(x)\to e^{-\frac {1}{2} x \left (a+\sqrt {b} x\right )} \left (c_1 \operatorname {HermiteH}\left (\frac {-a^2-4 \left (c+\sqrt {b}\right )}{8 \sqrt {b}},\sqrt [4]{b} x\right )+c_2 \operatorname {Hypergeometric1F1}\left (\frac {a^2+4 \left (c+\sqrt {b}\right )}{16 \sqrt {b}},\frac {1}{2},\sqrt {b} x^2\right )\right ) \]