29.19.17 problem 530

Internal problem ID [5126]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 19
Problem number : 530
Date solved : Sunday, March 30, 2025 at 06:42:38 AM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 48

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

\[ \frac {\arctan \left (\frac {y}{\sqrt {x^{2}-y^{2}}}\right ) x +\ln \left (x \right ) x -c_1 x -\sqrt {x^{2}-y^{2}}}{x} = 0 \]

✓ Mathematica. Time used: 0.343 (sec). Leaf size: 109

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

\[ \text {Solve}\left [\frac {2 \sqrt {\frac {y(x)}{x}-1} \text {arctanh}\left (\frac {1}{\sqrt {\frac {\frac {y(x)}{x}-1}{\frac {y(x)}{x}+1}}}\right )+\left (\frac {y(x)}{x}-1\right ) \sqrt {\frac {y(x)}{x}+1}}{\sqrt {\frac {\frac {y(x)}{x}-1}{\frac {y(x)}{x}+1}} \sqrt {\frac {y(x)}{x}+1}}=c_1-i \log (x),y(x)\right ] \]

✓ Sympy. Time used: 3.956 (sec). Leaf size: 24

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

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