83.5.3 problem 1 (c)

Internal problem ID [21938]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Chapter III. First order differential equations of the first degree. Ex. VI at page 47
Problem number : 1 (c)
Date solved : Thursday, October 02, 2025 at 08:16:43 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 22

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

\[ y = \tan \left (\operatorname {RootOf}\left (-\textit {\_Z} +\ln \left (\tan \left (\textit {\_Z} \right )\right )+2 \ln \left (x \right )+2 c_1 \right )\right ) x \]

✓ Mathematica. Time used: 0.076 (sec). Leaf size: 28

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

\[ \text {Solve}\left [\log \left (\frac {y(x)}{x}\right )-\arctan \left (\frac {y(x)}{x}\right )=-2 \log (x)+c_1,y(x)\right ] \]

✓ Sympy. Time used: 0.607 (sec). Leaf size: 22

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

\[ \log {\left (x \right )} = C_{1} - \log {\left (\sqrt {\frac {y{\left (x \right )}}{x}} \right )} + \frac {\operatorname {atan}{\left (\frac {y{\left (x \right )}}{x} \right )}}{2} \]