23.2.229 problem 235

Internal problem ID [5584]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 2. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF SECOND OR HIGHER DEGREE, page 278
Problem number : 235
Date solved : Tuesday, September 30, 2025 at 01:00:38 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.050 (sec). Leaf size: 115

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

\begin{align*} y &= 0 \\ a \,\operatorname {csgn}\left (a \right ) \ln \left (\frac {a \left (\operatorname {csgn}\left (a \right ) \sqrt {a^{2}-y^{2}}+a \right )}{y}\right )+a \,\operatorname {csgn}\left (a \right ) \ln \left (2\right )-\sqrt {a^{2}-y^{2}}-c_1 +x &= 0 \\ -a \,\operatorname {csgn}\left (a \right ) \ln \left (\frac {a \left (\operatorname {csgn}\left (a \right ) \sqrt {a^{2}-y^{2}}+a \right )}{y}\right )-a \,\operatorname {csgn}\left (a \right ) \ln \left (2\right )+\sqrt {a^{2}-y^{2}}-c_1 +x &= 0 \\ \end{align*}

✓ Mathematica. Time used: 0.202 (sec). Leaf size: 102

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

\begin{align*} y(x)&\to \text {InverseFunction}\left [\sqrt {a^2-\text {$\#$1}^2}-a \text {arctanh}\left (\frac {\sqrt {a^2-\text {$\#$1}^2}}{a}\right )\&\right ][-x+c_1]\\ y(x)&\to \text {InverseFunction}\left [\sqrt {a^2-\text {$\#$1}^2}-a \text {arctanh}\left (\frac {\sqrt {a^2-\text {$\#$1}^2}}{a}\right )\&\right ][x+c_1]\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 3.093 (sec). Leaf size: 41

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

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