Question

In Problems 1–22 solve the given differential equation by separation

of variables.

$\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{x}}\mathbf{+}\mathbf{2}\mathit{x}{\mathit{y}}^{\mathbf{2}}\mathbf{=}\mathbf{0}$

Short answer

The solution of the given differential equation is.$y=\frac{1}{x^2+C}$

Step-by-step solution

Step 1: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 integrate

The given equation is,$\frac{dy}{dx}+2x{y}^2=0$.

Separate the variables,

$\begin{array}{l}\frac{dy}{y^2}=-2xdx\\ y^{-2}dy=-2xdx\end{array}$

Integrate both sides,

$\begin{array}{c}\int y^{-2}dy=-\int 2xdx\\ -\frac{1}{y}=-x^2-C_1\end{array}$

Solve for y,

$\frac{1}{x^2-C_1}=y$

Substitute,$C=-C_1$

$y=\frac{1}{x^2+C}$

Hence, the solution of the given differential equation is.$\mathit{y}\mathbf{=}\frac{\mathbf{1}}{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}\mathbf{C}}$