Question

y’’-4y’+4y=0

Answer 1

what is the the auxillary equation?

Answer 2

auxiliary *

Answer 3

sometimes called the characteristic equation

Answer 4

Do you know what i'm talking about?

Answer 5

differential equations of first order

Answer 6

hmm, this one is not first order

Answer 7

the y'' term makes it a second order DE

Answer 8

I think it's 2 line

Answer 9

we have a second order differential equation of this form
\[ay''+by'+cy=0\]
if we let
\[y=e^{mx}\]\[y'=me^{mx}\]\[y''=m^2e^{mx}\]
and substitute
\[am^2e^{mx}+bme^{mx}+ce^{mx}=0\]
\[(am^2+bm+c)e^{mx}=0\]
We have a product of two terms
If the product of any two terms equals zero , one of the terms must be zero,
e^mx is not zero

we arrive at

\[\large am^2+bm+c=0\]
the auxiliary equation

Answer 10

but you dont have to go through this rigmarole every time

just jump from

\[ay''+by'+cy=0\]\[\qquad\Downarrow\qquad\Downarrow\]
to the auxiliary equation \[am^2+bm+c=0\]

Answer 11

aham ok thank you very much

Answer 12

:)

Answer 13

have you found your auxiliary equation ?

Answer 14

next step is to solve for m using the quadratic formula

Answer 15

\[am^2+bm+c=0\]
\[m_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\]