Find the solution of the differential equation: dy/dx = (xy^2 + x)/y. There is no need to rearrange the solution to be in terms of y.

This is a separable differential equation, so the first step is to separate the two variables. Factorise out the x from the top bracket so that the equation becomes: dy/dx = x(y^2+1)/yThe next step is to multiply the dx to the right hand side, and move the (y^2+1)/y to the left hand side, making it now look like: y/(y^2+1) dy = x dxFrom there, all that has to be done is to integrate both sides, the left hand side on closer inspection is of the form f’(y)/f(y), with a factor of 2 missing, so it becomes 1/2 log(y^2+1) while the right hand side is a straightforward integral and becomes 1/2 x^2. Putting all of this together and not forgetting the constant of integration, the overall solution is:log(y^2+1) = x^2 + c (Both sides have been multiplied by 2)