26.1.6 problem First order with homogeneous Coefficients. Exercise 7.7, page 61

Internal problem ID [6911]
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 7.7, page 61
Date solved : Tuesday, September 30, 2025 at 04:04:52 PM
CAS classification : [[_homogeneous, `class G`], _dAlembert]

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 32

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

\[ \frac {-c_1 x y+y+\sqrt {y^{2}-x^{2}}}{x y} = 0 \]

✓ Mathematica. Time used: 1.252 (sec). Leaf size: 107

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

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

✓ Sympy. Time used: 1.721 (sec). Leaf size: 32

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

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