80.6.8 problem 9 (b)

Internal problem ID [18501]
Book : Elementary Differential Equations. By Thornton C. Fry. D Van Nostrand. NY. First Edition (1929)
Section : Chapter IV. Methods of solution: First order equations. section 33. Problems at page 91
Problem number : 9 (b)
Date solved : Monday, March 31, 2025 at 05:38:25 PM
CAS classification : [_separable]

\begin{align*} \sqrt {-u^{2}+1}\, v^{\prime }&=2 u \sqrt {1-v^{2}} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 32

ode:=(-u^2+1)^(1/2)*diff(v(u),u) = 2*u*(1-v(u)^2)^(1/2); 
dsolve(ode,v(u), singsol=all);
 

\[ v = \sin \left (\frac {2 c_1 \sqrt {-u^{2}+1}+2 u^{2}-2}{\sqrt {-u^{2}+1}}\right ) \]

✓ Mathematica. Time used: 0.257 (sec). Leaf size: 44

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

\begin{align*} v(u)\to -\sin \left (2 \sqrt {1-u^2}-c_1\right ) \\ v(u)\to -1 \\ v(u)\to 1 \\ v(u)\to \text {Interval}[\{-1,1\}] \\ \end{align*}

✓ Sympy. Time used: 0.342 (sec). Leaf size: 15

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

\[ v{\left (u \right )} = \sin {\left (C_{1} - 2 \sqrt {1 - u^{2}} \right )} \]