60.3.267 problem 1283

Internal problem ID [11263]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1283
Date solved : Wednesday, March 05, 2025 at 02:04:47 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 4 x^{2} y^{\prime \prime }+4 x^{2} \ln \left (x \right ) y^{\prime }+\left (x^{2} \ln \left (x \right )^{2}+2 x -8\right ) y-4 x^{2} \sqrt {{\mathrm e}^{x} x^{-x}}&=0 \end{align*}

✓ Maple. Time used: 0.016 (sec). Leaf size: 48

ode:=4*x^2*diff(diff(y(x),x),x)+4*x^2*ln(x)*diff(y(x),x)+(x^2*ln(x)^2+2*x-8)*y(x)-4*x^2*(exp(x)/(x^x))^(1/2) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {x^{2} \left (\ln \left (x \right )-\frac {1}{3}\right ) \sqrt {x^{-x} {\mathrm e}^{x}}}{3}+{\mathrm e}^{\frac {x}{2}} \left (c_{1} x^{-\frac {x}{2}+2}+c_{2} x^{-\frac {x}{2}-1}\right ) \]

✓ Mathematica. Time used: 0.106 (sec). Leaf size: 89

ode=-4*x^2*Sqrt[E^x/x^x] + (-8 + 2*x + x^2*Log[x]^2)*y[x] + 4*x^2*Log[x]*D[y[x],x] + 4*x^2*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_1 e^{x/2} x^{-\frac {x}{2}-1}+\frac {1}{3} c_2 e^{x/2} x^{2-\frac {x}{2}}-\frac {1}{9} \sqrt {e^x x^{-x}} x^2+\frac {1}{3} \sqrt {e^x x^{-x}} x^2 \log (x) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-4*x**2*sqrt(exp(x)/x**x) + 4*x**2*log(x)*Derivative(y(x), x) + 4*x**2*Derivative(y(x), (x, 2)) + (x**2*log(x)**2 + 2*x - 8)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (4*x**2*(sqrt(exp(x)/x**x) - Derivative(y(x), (x, 2))) - x**2*y(x)*log(x)**2 - 2*x*y(x) + 8*y(x))/(4*x**2*log(x)) cannot be solved by the factorable group method