Question

I need help finding the general solution for the differential equation 6y''' +7y'' -y' -2y = 0, with x as the independent variable

Answer 1

seems to be a bit evil to me, I guess you have tried 6r^3+7r^2-r-2=0 already?

Answer 2

yes, i have. does p(m/n) sound familiar to you?

Answer 3

well, r or gamma, doesnt matter

Answer 4

not in that form, but it's easy to find the solutions of the cubic polynomial above.

Answer 5

Were you refering to the rational root theorem? Or are you talking about your third order homogenous differential equation?

Answer 6

6r^3+7r^2-r-2=0 has three roots. We can verify that -1 is a root based on inspection or by using a calculator. Then you can divide 6r^3+7r^2-r-2=0 by (x+1) using long division or synthetic division. You will end up with a quadratic polynomial then...

Answer 7

rational root theorum.

Answer 8

yes as @jheidelk did point already, you can find all of the roots above by either synthetic division, or rational root theorem, then you have all the solutions.

I can't LaTeX at this computer but they are of the form e^(rt)

Answer 9

the general solution is a linear superposition of all the solutions you obtain.

Answer 10

i mean, the answer is in front of me, i just dont understand how the second and third e^(rt). The book says the second one is e^(-2x/3) and the third is e^(x/2)

Answer 11

http://www.wolframalpha.com/input/?i=6r%5E3%2B7r%5E2-r-2%3D0

Answer 12

Do you know how to solve a cubic polynomial as above?

Answer 13

no

Answer 14

Okay, so when you divide 6r^3+7r^2-r-2=0 by (x+1), you get 6r^2+r-2. That factors into (2r-1) and (3r+2)

Answer 15

Well I think you do since you refered to the rational root theorem already, the solutions of it are identical to the exponent of the solution of your homogenous linear differential equation.

Answer 16

So there are three roots, -1, 1/2, -2/3

Answer 17

....my god.... I hate factoring

Answer 18

then use the quadratic equation? (-:

Answer 19

alpha wolf definitely showed me the other form i needed to get those roots