Question

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

of variables.

$\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{x}}\mathbf{=}{\mathbf{(}\mathbf{x}\mathbf{+}\mathbf{1}\mathbf{)}}^{\mathbf{2}}$

Short answer

The solution of the given differential equation is.$y=\frac{1}{3}{(x+1)}^3+C$

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,$\frac{dy}{dx}={(x+1)}^2$.

Separate the variables,$dy={(x+1)}^{2}dx$.

Integrate both sides,

$\int dy=\int {(x+1)}^{2}dx$

Do the integration, we get

$y=\frac{1}{3}{(x+1)}^3+C$

Hence, the solution of the given differential equation is .$\mathit{y}\mathbf{=}\frac{\mathbf{1}}{\mathbf{3}}{\mathbf{(}\mathbf{x}\mathbf{+}\mathbf{1}\mathbf{)}}^{\mathbf{3}}\mathbf{+}\mathit{C}$