61.26.3 problem 3

Internal problem ID [12424]
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 Equations Containing Power Functions. page 213
Problem number : 3
Date solved : Monday, March 31, 2025 at 05:34:03 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 22

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

\[ y = {\mathrm e}^{\frac {a \,x^{2}}{2}} \left (c_2 \,\operatorname {erf}\left (\sqrt {a}\, x \right )+c_1 \right ) \]

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 43

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

\[ y(x)\to c_1 \operatorname {ParabolicCylinderD}\left (-1,\sqrt {2} \sqrt {a} x\right )+c_2 \operatorname {ParabolicCylinderD}\left (0,i \sqrt {2} \sqrt {a} x\right ) \]

✗ Sympy

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

False