32.6.19 problem Exercise 12.19, page 103

Internal problem ID [5884]
Book : Ordinary Differential Equations, By Tenenbaum and Pollard. Dover, NY 1963
Section : Chapter 2. Special types of differential equations of the first kind. Lesson 12, Miscellaneous Methods
Problem number : Exercise 12.19, page 103
Date solved : Sunday, March 30, 2025 at 10:22:48 AM
CAS classification : [NONE]

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

✓ Maple. Time used: 0.112 (sec). Leaf size: 34

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

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

✓ Mathematica. Time used: 1.757 (sec). Leaf size: 44

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

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

✗ Sympy

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

Timed Out