Question

Verify that the vector X is a solution of the given system:
dx/dt = 3x - 4y
dy / dt = 4x - 7y
\[x = \left(\begin{matrix}1 \\ 2\end{matrix}\right) e ^{-5t}\]

Answer 1

this is a verification/substitution problem , would you like help :)

Answer 2

Yeah I would. I know the rule states that if the answer can be derived, than it is a solution vector, but I'm not too sure how to get to there.

Answer 3

That I get, and we can multiply to the initial system, right?

Answer 4

$$
{ \bf \vec x } = \left(\begin{matrix}x \\ y\end{matrix}\right) = \left(\begin{matrix}1\cdot e ^{-5t} \\ 2 \cdot e ^{-5t}\end{matrix}\right)
$$

Answer 5

yes, then we set
x(t) = e^{-5t}
y(t) = 2e^{-5t}

find dx/dt,
dy/dt
then plug in

Answer 6

yeah so matrix multiplication, I would get 3e^-5t - 8 (e^-5t) for dx/dt and
4e^-5t -14e^-5t for dy/dt right?

Answer 7

I got
x(t) = e^{-5t}
dx/dt = -5e^{-5t}

y(t) = 2e^{-5t}
dy/dt = 2(-5)e^{-5t}

Answer 8

ok one moment, two steps behind you

Answer 9

oh wait you derived it. my bad. I thought you did a matrix multiplication. Yeah I get the samething when I derive

Answer 10

ok :)

Answer 11

why would you derive right away and not make a matrix multiplication?

Answer 12

this is a checking problem. you want to check that
x(t) = e^{-5t}, y(t) = 2e^{-5t}

satisfy the system of equations :
dx/dt = 3x - 4y
dy / dt = 4x - 7y

Answer 13

the x(t), y(t) are the components of the vector solution x

Answer 14

Oh I see, so because my system are derivatived, I need to do the same to my solutions?

Answer 15

right

Answer 16

for example i get
dx/dt = -5 e^{-5t}

3x - 4y = 3e^{-5t} - 4 (2e^{-5t}) = (3-8) e^{-5t} =-5 e^{-5t}

Therefore dx/dt = 3x-4y \( \Large \checkmark \)

Answer 17

ohhh I see ,that makes sense, thank you ^^ So the derivative of the solution must be equal to the solution space?

Answer 18

to the system* my bad

Answer 19

right :)

Answer 20

Are you studying linear algebra?

Answer 21

No differential equations. I already studied linear algebra :)

Answer 22

ok :)