# For the differential equation $xy\large\frac{dy}{dx}=$$(x+2)(y+2), find the solution curve passing through the point (1,-1). \begin{array}{1 1}y - x +2 = log(x^2)(y+2)^2 \\ y+ x +2 = log(x^2)(y-2)^2 \\ y - x - 2 = log(x^2)(y-2)^2 \\ y + x +2 = log(x^2)(y+2)^2 \end{array} ## 1 Answer Toolbox: • Whenever a function occurs in the form \int\large\frac{x}{x+a}, then we can integrate by adding and subtracting a to the numerator. Step 1: given xy\large\frac{dy}{dx}$$= (x+2)(y+2)$
On rearranging we get $\large\frac{ydy}{y+2} =\frac{ (x+2)dx}{x}$
Using the information in the tool box,
$\large\frac{(y+2-2)dy}{y+2 }= $$dx + \large\frac{2dx}{x} \large\frac{(y+2)dy}{y+2 }-\frac{ 2dy}{y+2}$$ = dx + \large\frac{2dx}{x}$
Step 2:
On integrating both sides we get,