21. Solve dy/dx=x\sqrt{1-y^2} be separable method

21. Solve $\frac{dy}{dx}=x\sqrt{1-y^2}$ be separable method

To solve the differential equation $\frac{dy}{dx}=x\sqrt{1-y^2}$ by separable method, we need to rearrange it so that all the $y$ terms are on one side and all the $x$ terms are on the other side. Then, we can integrate both sides to obtain the solution.

First, we can divide both sides by $\sqrt{1-y^2}$ to get

$\frac{dy}{\sqrt{1-y^2}} = x dx$

Now, we can integrate both sides with respect to their respective variables:

$\int \frac{dy}{\sqrt{1-y^2}} = \int x dx$

To integrate the left-hand side, we can use the substitution $y=\sin u$ , which gives us $dy=\cos u du$ and $\sqrt{1-y^2} = \cos u$ . Substituting these expressions and using the identity $\int \cos^2 u du = \frac{1}{2}(u+\sin u + C)$ , where $C$ is the constant of integration, we get: