60.3.111 problem 1125

Internal problem ID [11107]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1125
Date solved : Wednesday, March 05, 2025 at 01:42:21 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} x y^{\prime \prime }+\left (4 x^{2}-1\right ) y^{\prime }-4 x^{3} y-4 x^{5}&=0 \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 36

ode:=x*diff(diff(y(x),x),x)+(4*x^2-1)*diff(y(x),x)-4*x^3*y(x)-4*x^5 = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{x^{2} \left (\sqrt {2}-1\right )} c_{2} +{\mathrm e}^{-x^{2} \left (1+\sqrt {2}\right )} c_{1} -x^{2}-2 \]

✓ Mathematica. Time used: 0.311 (sec). Leaf size: 45

ode=-4*x^5 - 4*x^3*y[x] + (-1 + 4*x^2)*D[y[x],x] + x*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to -x^2+c_1 e^{\left (\sqrt {2}-1\right ) x^2}+c_2 e^{-\left (\left (1+\sqrt {2}\right ) x^2\right )}-2 \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-4*x**5 - 4*x**3*y(x) + x*Derivative(y(x), (x, 2)) + (4*x**2 - 1)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -x*(4*x**4 + 4*x**2*y(x) - Derivative(y(x), (x, 2)))/(4*x**2 - 1) + Derivative(y(x), x) cannot be solved by the factorable group method