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:

    \[\int \frac{dy}{\sqrt{1-y^2}} = \int \frac{\cos u}{\cos u} du = u + C = \sin^{-1} y + C\]

To integrate the right-hand side, we can simply use the power rule:

    \[\int x dx = \frac{x^2}{2} + C\]

Therefore, the general solution to the differential equation is:

    \[\sin^{-1} y = \frac{x^2}{2} + C\]

    \[y = sin\left(\frac{x^2}{2} + C\right)\]

where C is an arbitrary constant of integration.

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