Question

how to solve this.
find the complementary solution, Yc to the homogeneous ODE
d^4y/dx^4 - 4 d^3y/dx^3 + 7 d^2y/dy^2 - 6 dy/dx + 2y = 0

Answer 1

first, write it in a notation that is flattering

Answer 2

y^4' - y'''+ 7y'' - 6y' + 2y = 0

Answer 3

then root it out like any other equation

Answer 4

\[r^4 - r^3 +7r^2-6r+2=0\]
such that:\[y=c_1e^{r_1x}+c_2e^{r_2x}+c_3e^{r_3x}+c_4e^{r_4x}\]

Answer 5

start making some educated guesses for a root :)

1,-1,2,-2 seem to be the best options for a rational solution

Answer 6

how you get that root

Answer 7

what root? i havent found a root yet, just some educated guesses at what it might be to try out before things get hairy

Answer 8

1 -1 7 -6 2
0 -2 6
1 -3 13 ...... the rational roots doesnt seem to wanna play nice

Answer 9

im missing a 4 in the equation .... thanks to the hideous way it was entered

Answer 10

\[r^4 - 4r^3 +7r^2-6r+2=0\]

Answer 11

now 1 IS a root :)


when r=1

1 - 4 +7- 6+2 = 10-10 = 0

Answer 12

1 -4 +7 -6 +2
0 1 -3 4
1 -3 4 -2
..................

\[(r-1)(r^3-3r^2+4r-2)=0\]

Answer 13

r=1 again; 1-3+4-2 = 5-5 = 0

Answer 14

1 -3 +4 -2
0 1 -2
1 -2 2
--------

\[(r-1)^2(r^2-2r+2)=0\]

Answer 15

the rest are complex

r^2-2r = -2

(r-1)^2 = -1

r = 1+- i

sooo, 2 repeats and a complex

Answer 16

\[y=c_1e^x+c_2x\ e^x+e^x(c_3cos(x)+c_4sin(x))\]

Answer 17

now i can answer any questions since we know what the end result should be :)

Answer 18

is that the final answer

Answer 19

dunno, feel free to dbl chk my math to make sure ...

Answer 20

thanks

Answer 21

yw