15.4.23 problem 24

Internal problem ID [2936]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 8, page 34
Problem number : 24
Date solved : Sunday, March 30, 2025 at 01:00:09 AM
CAS classification : [_exact]

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

✓ Maple. Time used: 0.013 (sec). Leaf size: 39

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

\begin{align*} y &= \sqrt {-x^{2}+{\mathrm e}^{-\frac {c_1}{x}}} \\ y &= -\sqrt {-x^{2}+{\mathrm e}^{-\frac {c_1}{x}}} \\ \end{align*}

✓ Mathematica. Time used: 0.945 (sec). Leaf size: 47

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

\begin{align*} y(x)\to -\sqrt {-x^2+e^{\frac {c_1}{x}}} \\ y(x)\to \sqrt {-x^2+e^{\frac {c_1}{x}}} \\ \end{align*}

✗ Sympy

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

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