75.4.11 problem 56

Internal problem ID [16713]
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 : Tuesday, January 28, 2025 at 09:19:12 AM
CAS classification : [_separable]

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

✓ Solution by Maple

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

dsolve(2*x*sqrt(1-y(x)^2)=diff(y(x),x)*(1+x^2),y(x), singsol=all)
 

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

✓ Solution by Mathematica

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

DSolve[2*x*Sqrt[1-y[x]^2]==D[y[x],x]*(1+x^2),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*}