Science & engineering math: system of differential equations

chatterbug219

1. The problem statement, all variables and given/known data

Solve the system of differential equations:
y'(t) + z(t) = t
y"(t) - z(t) = e^-t
Subject to y(0) = 3, y'(0) = -2, and z(0) = 0

2. Relevant equations

My professor did an example in class that was much simpler and solved it using Kramer's rule.

3. The attempt at a solution
I don't know how to start it. I thought about rearranging the equations so that one was equal to y'(t) and the other was equal to z(t), but I'm not sure that would work...

ehild

What about adding the equations so as z(t) cancels?

ehild

OldEngr63

I would definitely start off as ehild has suggested.

HallsofIvy

There is, however, a problem with the entire exercise. Doing as ehild suggests gives you a second order differential equation in y only which you can solve and then use the initial conditions to give a specific solution for y. But there is no derivative of z in these equations- once you know y, z is fixed and you have no constant to choose to make z(0)= 0. Was one or both of those "z"s supposed to be z'? If not then any one of the three conditions, y(0)= 3, y'(0)= -2, z(0)=n 0, can be dropped to give a solution but there is not y, z, satisfying the equations and all three of the conditions.

chatterbug219

Oh my gosh yes there was supposed to be a z' in the first equation
So it is: y'(t) + z'(t) = t
The second equation is correct though, so sorry for any confusion!!

ehild

Then you can integrate the first equation and add to the second.

ehild