75.7.5 problem 179

Internal problem ID [16727]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 7, Total differential equations. The integrating factor. Exercises page 61
Problem number : 179
Date solved : Monday, March 31, 2025 at 03:12:13 PM
CAS classification : [[_homogeneous, `class D`], _exact, _rational]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 65

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

\begin{align*} y &= -\frac {\left (-x^{2}+\sqrt {x^{4}+4 c_1 \,x^{2}+4 c_1^{2}+4}-2 c_1 \right ) x}{2} \\ y &= \frac {\left (x^{2}+2 c_1 +\sqrt {x^{4}+4 c_1 \,x^{2}+4 c_1^{2}+4}\right ) x}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.416 (sec). Leaf size: 78

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

\begin{align*} y(x)\to \frac {1}{2} x \left (x^2-\sqrt {x^4+2 c_1 x^2+4+c_1{}^2}+c_1\right ) \\ y(x)\to \frac {1}{2} x \left (x^2+\sqrt {x^4+2 c_1 x^2+4+c_1{}^2}+c_1\right ) \\ y(x)\to 0 \\ \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (2*x**3*y(x) + x**2 + y(x)**2)*y(x)/(x*(x**2 + y(x)**2)) cannot be solved by the factorable group method