82.12.11 problem Ex. 11

Internal problem ID [18705]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter II. Equations of the first order and of the first degree. Examples on chapter II at page 29
Problem number : Ex. 11
Date solved : Monday, March 31, 2025 at 05:59:34 PM
CAS classification : [_exact, _rational]

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

✓ Maple. Time used: 0.012 (sec). Leaf size: 257

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

\begin{align*} y &= -\frac {\sqrt {-4 b^{2}+4 x^{2}-2 \sqrt {8 x^{4}+\left (-8 a^{2}-8 b^{2}\right ) x^{2}+2 a^{4}+4 a^{2} b^{2}+2 b^{4}+16 c_1}}}{2} \\ y &= \frac {\sqrt {-4 b^{2}+4 x^{2}-2 \sqrt {8 x^{4}+\left (-8 a^{2}-8 b^{2}\right ) x^{2}+2 a^{4}+4 a^{2} b^{2}+2 b^{4}+16 c_1}}}{2} \\ y &= -\frac {\sqrt {-4 b^{2}+4 x^{2}+2 \sqrt {8 x^{4}+\left (-8 a^{2}-8 b^{2}\right ) x^{2}+2 a^{4}+4 a^{2} b^{2}+2 b^{4}+16 c_1}}}{2} \\ y &= \frac {\sqrt {-4 b^{2}+4 x^{2}+2 \sqrt {8 x^{4}+\left (-8 a^{2}-8 b^{2}\right ) x^{2}+2 a^{4}+4 a^{2} b^{2}+2 b^{4}+16 c_1}}}{2} \\ \end{align*}

✓ Mathematica. Time used: 2.633 (sec). Leaf size: 209

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

\begin{align*} y(x)\to -\sqrt {-\sqrt {-2 a^2 x^2+b^4-2 b^2 x^2+2 x^4+4 c_1}-b^2+x^2} \\ y(x)\to \sqrt {-\sqrt {-2 a^2 x^2+b^4-2 b^2 x^2+2 x^4+4 c_1}-b^2+x^2} \\ y(x)\to -\sqrt {\sqrt {-2 a^2 x^2+b^4-2 b^2 x^2+2 x^4+4 c_1}-b^2+x^2} \\ y(x)\to \sqrt {\sqrt {-2 a^2 x^2+b^4-2 b^2 x^2+2 x^4+4 c_1}-b^2+x^2} \\ \end{align*}

✗ Sympy

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

Timed Out