Question

In Problems 1–22, solve the given differential equation by separation of variables.

$\frac{\text{dy}}{\text{dx}}\text{= x}\sqrt{{\text{1 -y}}^{\text{5}}}$

Short answer

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

Step-by-step solution

Definition of "separable equation"

A first-order differential equation of the form $\frac{dy}{dx}=g\left(x\right)h\left(y\right)$is said to be separable or to have separable variables.

Separate the variables and integration

Separate the variables.

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

Integrate both sides

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

Do the integration.

$$\begin{gathered} {\sin }^{-1}y=\frac{x^2}{2}+C \\ y=\sin \left(\frac{x^2}{2}+C\right) \end{gathered}$$

Hence, the solution is$\text{y = sin}\left(\frac{{\text{x}}^{\text{2}}}{\text{2}}\text{+ C}\right)$