Solving the Initial value problem

[chen0000]

Homework Statement

dx/ dt = x + y
dy/ dt = x + y + e^t
x(0) = 0 y(0) = 1.

Homework Equations

The Attempt at a Solution

x'' = x' + y' = x' + x + y + e^t
x'' - 2x' = 0
y = x' - x

 

[tiny-tim]

welcome to pf!

hi chen0000! welcome to pf!

dx/ dt = x + y
dy/ dt = x + y + e^t
x(0) = 0 y(0) = 1.

the obvious way to start would be by changing one of the variables to x+y

 

[chen0000]

well I'm not sure if we're supposed to do it like this but I remember something about
x" = x' + y' = x' + x + y + e^t
and then we have x'' - 2x' = 0 and y = x' - x, but that's for homogeneous, and i don't know where to go from there.

 

[tiny-tim]

… then we have x'' - 2x' = 0 and y = x' - x

ok, solve x'' - 2x' = e^t first …

what do you get? ​

 

[chen0000]

x'' - 2x' = e^t
X(t) = Ae^t
(A-2A)e^t = e^t
-A = 1
A = -1
X(t) = - e^t ?

 

[tiny-tim]

hi chen0000!

(just got up :zzz: …)

x'' - 2x' = e^t
X(t) = Ae^t
(A-2A)e^t = e^t
-A = 1
A = -1
X(t) = - e^t ?

(try using the X² icon just above the Reply box )

yes, that's a good particular solution …

now find the homogenous solution (ie the general solution to the homogenous equation x'' - 2x = 0) to add to that ​