Solve ylnx dx/dy =({y+1}/{x})^2 by separation of variables

The given differential equation is:

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

To solve this equation by separation of variables, we can first move the dx term to the left-hand side and the y term to the right-hand side, giving:

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

Now, we can integrate both sides. On the left-hand side, the integral becomes as.

(3)   \begin{equation*} \int x^2ln(x){dx} = \frac{1}{3}x^3\ln \left(x\right)-\frac{x^3}{9}+ C_1. \end{equation*}

Where C_1 is the constant of integration.

On the right-hand side, we can integrate, so the integral becomes:

(4)   \begin{equation*} \int  \frac{({y+1})^2}{y}dy =\frac{y^2}{2}+2y+\ln \left(y\right) \end{equation*}

where C_2 is another constant of integration.

Putting the two integrals together, we have:

(5)   \begin{equation*} \frac{y^2}{2}+2y+\ln \left(y\right)=\frac{1}{3}x^3\ln \left(x\right)-\frac{x^3}{9}+C_1  \end{equation*}

where C_1 is the constant of integration.

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