60.1.188 problem 191

Internal problem ID [10202]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 191
Date solved : Sunday, March 30, 2025 at 03:30:11 PM
CAS classification : [_separable]

\begin{align*} \sqrt {-x^{2}+1}\, y^{\prime }-y \sqrt {y^{2}-1}&=0 \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 16

ode:=(-x^2+1)^(1/2)*diff(y(x),x)-y(x)*(-1+y(x)^2)^(1/2) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ \arcsin \left (x \right )+\arctan \left (\frac {1}{\sqrt {y^{2}-1}}\right )+c_1 = 0 \]

✓ Mathematica. Time used: 1.112 (sec). Leaf size: 50

ode=Sqrt[1-x^2]*D[y[x],x] - y[x]*Sqrt[y[x]^2-1]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to -\sqrt {\sec ^2(\arcsin (x)+c_1)} \\ y(x)\to \sqrt {\sec ^2(\arcsin (x)+c_1)} \\ y(x)\to -1 \\ y(x)\to 0 \\ y(x)\to 1 \\ \end{align*}

✓ Sympy. Time used: 0.574 (sec). Leaf size: 27

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

\[ \begin {cases} i \operatorname {acosh}{\left (\frac {1}{y{\left (x \right )}} \right )} & \text {for}\: \frac {1}{\left |{y^{2}{\left (x \right )}}\right |} > 1 \\- \operatorname {asin}{\left (\frac {1}{y{\left (x \right )}} \right )} & \text {otherwise} \end {cases} = C_{1} + \operatorname {asin}{\left (x \right )} \]