Give the general solution to the Ordinary Differential Equation: (dy/dx) + 2y/x = 3x+2

It can first be observed that this differential equation is linear, so we can solve it by multiplying the whole equation by the integrating factor. As there is no coefficient in front of the dy/dx term, we do not have to do anything to the equation before finding the integrating factor. The integrating factor is exp( integral (2/y) dx). From core maths we can solve this; the integral gives 2 ln (x). 2 ln (x) is equal to ln (x^2) using the rules of exponentials. exp ( ln(x^2) ) is x^2 and so the integrating factor is x^2.
We then multiply the equation by the integrating factor, x^2, to get x^2*(dy/dx) + 2xy = x^2(3x+2). We can recognise the left hand side as the product rule and so we can express the equation as: d(x^2y)/dx = 3x^3 + 2x^2. This is now separable and so we can use techniques learned from the maths a level to give: x^2y = 0.75x^4 + (2/3)x^3 + c, making sure to add the + c as it is the constant of integration. Dividing through by x^2 we get y = 0.75x^2 + 2x/3 + c/x^2. This is the general solution to the differential equation.