76.23.9 problem Ex. 10

Internal problem ID [20150]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter IV. Singular solutions. problems on chapter IV. page 49
Problem number : Ex. 10
Date solved : Thursday, October 02, 2025 at 05:33:18 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational, _Clairaut]

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

✓ Maple. Time used: 0.130 (sec). Leaf size: 82

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

\begin{align*} y &= \frac {\sqrt {a^{2}-x^{2}}\, b}{a} \\ y &= -\frac {\sqrt {a^{2}-x^{2}}\, b}{a} \\ y &= c_1 x -\sqrt {c_1^{2} a^{2}+b^{2}} \\ y &= c_1 x +\sqrt {c_1^{2} a^{2}+b^{2}} \\ \end{align*}

✓ Mathematica. Time used: 2.563 (sec). Leaf size: 419

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

\begin{align*} \text {Solve}\left [-\frac {\sqrt {a^2 \left (y(x)^2-b^2\right )} \arctan \left (\frac {\sqrt {y(x)^2-b^2}}{b}\right )}{b \sqrt {y(x)^2-b^2}}-\frac {2 \sqrt {y(x)^2-b^2} \sqrt {-a^2 \left (b^2-y(x)^2\right )} \arctan \left (\frac {b x \sqrt {y(x)^2-b^2}}{y(x) \left (\sqrt {a^2 \left (y(x)^2-b^2\right )}-\sqrt {a^2 \left (y(x)^2-b^2\right )+b^2 x^2}\right )+b^2 x}\right )}{b^3-b y(x)^2}=c_1,y(x)\right ]\\ \text {Solve}\left [\frac {\sqrt {a^2 \left (y(x)^2-b^2\right )} \arctan \left (\frac {\sqrt {y(x)^2-b^2}}{b}\right )}{b \sqrt {y(x)^2-b^2}}+\frac {2 \sqrt {y(x)^2-b^2} \sqrt {-a^2 \left (b^2-y(x)^2\right )} \arctan \left (\frac {b x \sqrt {y(x)^2-b^2}}{y(x) \left (\sqrt {a^2 \left (y(x)^2-b^2\right )+b^2 x^2}-\sqrt {a^2 \left (y(x)^2-b^2\right )}\right )+b^2 x}\right )}{b^3-b y(x)^2}=c_1,y(x)\right ]\\ y(x)&\to -\frac {b \sqrt {a^2-x^2}}{a}\\ y(x)&\to \frac {b \sqrt {a^2-x^2}}{a} \end{align*}

✗ Sympy

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

Timed Out