Ordinary Differential Equations

[Physter]

Homework Statement

http://img54.imageshack.us/img54/1663/scan0002li4.jpg

The Attempt at a Solution

Its been two years since I've seen differential equations and now I'm taking a more advanced course in the same area so our prof gave us a couple questions to see what we remember. Its not worth any marks or anything but he is suggesting we hand it into get an idea for where we are. I spoke to my TA and she wasn't exactly all that helpful so if someone can point me in the right direction, it'd be awesome. I just need some tips, please don't give me full solutions or anything of that sort. Thanks

a) I think this is a separable equation so my guess is I can rewrite the y' as dy/dx and then move the dx over to the other side of the equation and then differentiate both sides in order to solve for y.

b) Its a first order equation and I honestly have no idea what to do with it lol. Its not separable nor is it an exact equation so umm linear? Integrating factor? Something like that? :S

c) Its a second order inhomogeoneous equation so my final answer should be the sum of a homogenous solution and a particular solution. First I think i'd solve the homogeonous equation by guessing a solution. I've been told that if we have a cosx in the equation, I should guess the solution cosx; if its a sinx in the equation, an appropriate solution might be sinx; if its e^x, a solution might be e^x. What I don't understand is in our equation, we have cos2x so should I guess cos2x or simply go with cosx? But yes, after solving the homogenous equation, I'd look for a particular solution. That's something I don't remember at all so I kind of need a hand as to where I should start with that. And I know once I have the particular solution, I'd simply add it to the homogenous solution I first found and that would be my final answer.

d) Ok I know its a second order homogeneous equation but how do I know which method I should use to solve the equation? How do you know when to use constant coefficients, Euler's method or reduction of order?

e) Second order homogeoneous equation and again its the same problem as question d). How do I know which method to use?

f) Its a first order equation and I think it may be exact? I suppose I should check for exactness and then go from there?

 

[drpizza]

#2;
Have you learned the method of integrating factors?

 

[Physter]

The prof just mentioned it briefly but I don't really know how to apply it.

 

[dextercioby]

The first 2 don't require use of integrating factors. The first one is immediate by direct integration and the second can be integrated an inhomogenous linear 1-order ODE.

Daniel.