61.27.3 problem 13

Internal problem ID [12434]
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 : Monday, March 31, 2025 at 05:34:19 AM
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*}

✓ Maple. Time used: 0.050 (sec). Leaf size: 74

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

\[ y = {\mathrm e}^{-\frac {x \left (\sqrt {b}\, x +a \right )}{2}} x \left (\operatorname {KummerU}\left (\frac {a^{2}+12 \sqrt {b}+4 c}{16 \sqrt {b}}, \frac {3}{2}, \sqrt {b}\, x^{2}\right ) c_2 +\operatorname {KummerM}\left (\frac {a^{2}+12 \sqrt {b}+4 c}{16 \sqrt {b}}, \frac {3}{2}, \sqrt {b}\, x^{2}\right ) c_1 \right ) \]

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

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

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

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
y = Function("y") 
ode = Eq(a*Derivative(y(x), x) - (b*x**2 + c)*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False