# Thread: how to find a general solution (dfq)

1. ## 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.

2. Complimentary functions are solutions of the corresponding homogeneous equation $\displaystyle y'' + 4y = 0$. Generally, you find solutions to second-order linear constant coefficient ode's like this by assuming a solution of the form $\displaystyle y=Ae^{rx}.$ You plug that solution in, and you'll get an equation for $\displaystyle r$ that, in this case, will have complex solutions. Use Euler's formula to get the sines and cosines out of the exponentials.

Make sense?

3. 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

4. Ok, let me know how it goes.