3.24 problem 25

Internal problem ID [951]

Book: Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section: Chapter 2, First order equations. separable equations. Section 2.2 Page 52
Problem number: 25.
ODE order: 1.
ODE degree: 1.

CAS Maple gives this as type [_separable]

Solve \begin {gather*} \boxed {y^{\prime } \sqrt {-x^{2}+1}+\sqrt {1-y^{2}}=0} \end {gather*}

Solution by Maple

Time used: 0.006 (sec). Leaf size: 11

dsolve(diff(y(x),x)*sqrt(1-x^2)+sqrt(1-y(x)^2)=0,y(x), singsol=all)
 

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

Solution by Mathematica

Time used: 2.929 (sec). Leaf size: 114

DSolve[y'[x]*Sqrt[1-x^2]+Sqrt[1-y[x]^2]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\frac {\tan \left (\text {ArcTan}\left (\frac {x}{\sqrt {1-x^2}}\right )-c_1\right )}{\sqrt {\sec ^2\left (\text {ArcTan}\left (\frac {x}{\sqrt {1-x^2}}\right )-c_1\right )}} \\ y(x)\to \frac {\tan \left (\text {ArcTan}\left (\frac {x}{\sqrt {1-x^2}}\right )-c_1\right )}{\sqrt {\sec ^2\left (\text {ArcTan}\left (\frac {x}{\sqrt {1-x^2}}\right )-c_1\right )}} \\ y(x)\to -1 \\ y(x)\to 1 \\ \end{align*}