Given a differential equation (*), show that the solution curve is either a straight line or a parabola and find the equations of these curves.

This question is takes from the 2008 STEP I paper (Question 8). We are first given (dy/dx)^2 + x dy/dx + y = 0 where y is a function of x. Differentiating both sides of the equation with respect to x, gives: 2(dy/dx)(d2y/dx2) -dy/dx + x d2y/dx2 + dy/dx = 0 which gives 2(dy/dx)(d2y/dx2) + x d2y/dx2 = 0. Factorising out the second derivative term gives: (d2y/dx2) (2(dy/dx) - x) = 0. Since either factor must always equal 0 we have either d2y/dx2 = 0 or 2(dy/dx) - x = 0. For the first equation, integrating with respect to x twice gives: y = Ax + B (the equation of a straight line!). For the second, re-arranging for dy/dx = x/2 then integrating with respect to x gives y = 1/6 x^2 + C. Substituting these solutions (involving the arbitrary constant!) and their respective derivatives back into the original differential equation (*) (which we know we can do because the solution equation must satisfy the differential equation), gives us B = A^2 in the first case and C = 1/12 x^2 in the second.