61.26.5 problem 5

Internal problem ID [12426]
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 : 5
Date solved : Monday, March 31, 2025 at 05:34:06 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+a^{3} x \left (-a x +2\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 26

ode:=diff(diff(y(x),x),x)+a^3*x*(-a*x+2)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.37 (sec). Leaf size: 44

ode=D[y[x],{x,2}]+a^3*x*(2-a*x)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to e^{-\frac {1}{2} a x (a x-2)} \left (c_2 \int _1^xe^{a K[1] (a K[1]-2)}dK[1]+c_1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(a**3*x*(-a*x + 2)*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False