Thread: how to find a general solution (dfq)

ok. so i am given an equation.
y''+4y=12x
i am told that the a particular equation is : y=3x
i understand that, it makes the first function =0. i know how to do that.
but then i am given the complimentary function as y=c1cos(2x)+c2sin(2x)

so my question is, where does that come from? how could i have found that on my own? i have a test and i believe our professor mentioned that we wouldn't be given those as we have been in our homework sets, so i need to be capable of finding it just from
y''+4y=12x.

i don't know how though. i did figure the particular one out, but this i am stuck.

Complimentary functions are solutions of the corresponding homogeneous equation . Generally, you find solutions to second-order linear constant coefficient ode's like this by assuming a solution of the form You plug that solution in, and you'll get an equation for that, in this case, will have complex solutions. Use Euler's formula to get the sines and cosines out of the exponentials.

Make sense?

yeah that makes sense, thanks. i have to run off to work unfortunately but i'll practice with it when i get back. but i'm pretty sure i understand. thank you

Ok, let me know how it goes.