Sovle by separation of variables dy/dx= (x+1)^2

Sovle by separation of variables dy/dx= (x+1)^2.

Given differential equation is:

(1)   \begin{equation*} \frac{dy}{dx} = (x+1)^2 \end{equation*}

We can separate the variables as follows:

(2)   \begin{equation*} \frac{dy}{(x+1)^2} = dx \end{equation*}

Now, we can integrate both sides of the equation:

(3)   \begin{equation*} \int \frac{dy}{(x+1)^2} = \int dx \end{equation*}

Integrating the left-hand side requires the substitution u = x+1 and du/dx = 1, yielding:

(4)   \begin{equation*} \int \frac{dy}{(x+1)^2} = \int \frac{du}{u^2} = -\frac{1}{u} + C_1 \end{equation*}

where C_1 is the constant of integration.

For the right-hand side, we can simply integrate with respect to x to obtain:

(5)   \begin{equation*} \int dx = x + C_2 \end{equation*}

where C_2 is another constant of integration.

Substituting back u=x+1, we have:

(6)   \begin{equation*} \int \frac{dy}{(x+1)^2} = -\frac{1}{x+1} + C_1 = x + C_2 \end{equation*}

Simplifying and solving for y, we get:

(7)   \begin{equation*} y = \frac{1}{3}(x+1)^3 - x - C \end{equation*}

where C = C_1 - C_2 is the constant of integration.

Therefore, the solution to the differential equation \frac{dy}{dx} = (x+1)^2 is:

(8)   \begin{equation*} y = \frac{1}{3}(x+1)^3 - x + C \end{equation*}

where C is the constant of integration.

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