83.23.19 problem 19

Internal problem ID [19229]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter V. Singular solutions. Exercise V at page 76
Problem number : 19
Date solved : Monday, March 31, 2025 at 06:58:31 PM
CAS classification : [_quadrature]

\begin{align*} \left (1-y^{2}\right ) {y^{\prime }}^{2}&=1 \end{align*}

✓ Maple. Time used: 0.051 (sec). Leaf size: 46

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

\begin{align*} y &= \sin \left (\operatorname {RootOf}\left (\sin \left (\textit {\_Z} \right ) \operatorname {csgn}\left (\cos \left (\textit {\_Z} \right )\right ) \cos \left (\textit {\_Z} \right )+\textit {\_Z} +2 c_1 -2 x \right )\right ) \\ y &= \sin \left (\operatorname {RootOf}\left (-\sin \left (\textit {\_Z} \right ) \operatorname {csgn}\left (\cos \left (\textit {\_Z} \right )\right ) \cos \left (\textit {\_Z} \right )-\textit {\_Z} +2 c_1 -2 x \right )\right ) \\ \end{align*}

✓ Mathematica. Time used: 0.052 (sec). Leaf size: 69

ode=(1-y[x]^2)*D[y[x],x]^2==1; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 1.415 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((1 - y(x)**2)*Derivative(y(x), x)**2 - 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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