Combine all the integrated terms: \[\int \left(\frac{1}{2+x} + \frac{4}{x(2+x)} + \frac{4}{x^2(2+x)} ight) dx = \ln|2+x| + 2\ln|x| + 2\ln|2+x| - \frac{4}{x} - 4\ln|2+x| + C \] Simplify: \[= 2\ln|x| - \frac{4}{x} + C \] Step 4: Combine and Solve for y y Combine the results from both sides: \[\frac{y^2}{2} = 2\ln|x| - \frac{4}{x} + C \] Multiply through by 2 to solve for y 2 y 2 : \[y^2 = 4\ln|x| - \frac{8}{x} + C' \] This is the general solution to the differential equation