First a word of explanation about my circumstances, skip the italics if you're not interested: I have a (vitally important) exam on differential equations coming up in two weeks, and I have no institutional support (lectures, tutors, etc). It's not a subject I've studied in any depth before but I'm a quick learner and I have to try. I'm sure given enough time I could find the information I need without having to ask, but I don't have a lot of time, and with other exams also coming up I need to concentrate on those too. Forgive me if the questions are insultingly easy, I simply the lack the knowledge to answer them myself Some questions from a past exam paper, I don't need a full set of workings and answers, just enough so that I can guess the rest. Or just giving me links to pages which the explanations I need would be fantastic. 1. Solve each of the following equations given the initial conditions: i) y' - 3y = 0, with y(0) = 2 ii) y'' + y' -6y = 0, with y(0) = 0 and y'(0) = 1 iii) y'' + 2y' +y = 0, with y(0) = 2 and y'(0) = 0 iv) y'' - 2y' +2y = 0, with y(0) = 0 and y'(0) = 1 I'm not even sure what this is asking. Substituting the values into the equations, clearly in the case of i) y'(0) = 6 and in the others would give a value for y''(0) but is this a solution? 2. i) Consider the following differential equation for y(x): y'' + y = 1 - x a) Find the complementary function for this equation b) find the particular integral for this equation I don't know what these terms mean c) Write down the solution to this equation, given the initial conditions y(0) = 1 and y'(0) = 1 ii) Consider the following differential equation for y(x): y'' + y' = e^-x a) b) and c) as above. There are three more questions on the paper, but I think I suffer from the same basic lack of knowledge for all of them, and help with these could shed some light on the others. Sorry for not using the template or making a serious attempt at an answer, I will make one if someone can point me in the right direction.