75.4.11 problem 56

Internal problem ID [16634]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 4. Equations with variables separable and equations reducible to them. Exercises page 38
Problem number : 56
Date solved : Thursday, March 13, 2025 at 08:28:04 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 15

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

\[ y = \sin \left (\ln \left (x^{2}+1\right )+2 c_{1} \right ) \]

✓ Mathematica. Time used: 0.241 (sec). Leaf size: 33

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

\begin{align*} y(x)\to \sin \left (\log \left (x^2+1\right )+c_1\right ) \\ y(x)\to -1 \\ y(x)\to 1 \\ y(x)\to \text {Interval}[\{-1,1\}] \\ \end{align*}

✓ Sympy. Time used: 0.317 (sec). Leaf size: 12

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

\[ y{\left (x \right )} = \sin {\left (C_{1} + \log {\left (x^{2} + 1 \right )} \right )} \]