Science & engineering math: system of differential equations

[chatterbug219]

Homework Statement

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

Homework Equations

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

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 anyone 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]

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!

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

ehild