32.1.2 problem First order with homogeneous Coefficients. Exercise 7.3, page 61

Internal problem ID [5772]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 2. Special types of differential equations of the first kind. Lesson 7
Problem number : First order with homogeneous Coefficients. Exercise 7.3, page 61
Date solved : Sunday, March 30, 2025 at 10:14:16 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.007 (sec). Leaf size: 33

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

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

✓ Mathematica. Time used: 1.756 (sec). Leaf size: 69

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

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

✓ Sympy. Time used: 2.458 (sec). Leaf size: 17

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

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