69.1.18 problem 21

Internal problem ID [14092]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 21
Date solved : Wednesday, March 05, 2025 at 10:31:03 PM
CAS classification : [_separable]

\begin{align*} \left (x^{2}+1\right ) y^{\prime }-\sqrt {1-y^{2}}&=0 \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 9

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

\[ y = \sin \left (\arctan \left (x \right )+c_{1} \right ) \]

✓ Mathematica. Time used: 0.256 (sec). Leaf size: 34

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

\begin{align*} y(x)\to \sin \left (\int _1^x\frac {1}{K[1]^2+1}dK[1]+c_1\right ) \\ y(x)\to -1 \\ y(x)\to 1 \\ \end{align*}

✓ Sympy. Time used: 0.273 (sec). Leaf size: 8

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

\[ y{\left (x \right )} = \sin {\left (C_{1} + \operatorname {atan}{\left (x \right )} \right )} \]