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    ok i've tried searching all over google for diff.eq, but they all come up in the form y"+y'+y or y'+y, but nothing like y"+y' or y"+y (hope this isn't too confusing! :p: )

    do these kind of diff.eq have a name so i can search for it on google more specifically? or can anyone tell me what the hell the teacher's doing?! :lolwut: (just small parts of it would help majorly!!) thanks!
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    Basic differential equations will come up in the form ay'' + by' + cy = f(x), but you're perfectly welcome to set b = 0 or c = 0 in there to get the equations you were talking about. The one your teacher has solved for you is a different one entirely - the coefficients are no longer constant (i.e. they're things like x and x^2, which are obviously changing as x changes), which makes things quite difficult. Your teacher has obviously seen this type of equation before, because (s)he knew that making the substitution x = e^t was going to be a helpful thing to do. The equation is now in terms of y and t rather than y and x, and turns out to be much nicer. (Exponential functions like e^ax are frequently very useful, because they are eigenfunctions of the differential equation operator - that is, their derivatives are constant multiples of themselves.)

    This website goes through loads of different types of differential equations - it's not very 'basic', but it does show you how to solve them.

    If you need any more help with specific lines of that solution, let me know - but there's not really much I can do if you don't tell me where you're struggling.
 
 
 
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