# differential equation (twist to it)

Discussion in 'Homework Help' started by squirby, Sep 18, 2009.

1. ### squirby Thread Starter Member

Aug 21, 2009
15
0
hey guys. this is a diff. equation with a twist.

"For a system, it is given that the complementary solution is 8exp(-3t)u(t).
The particular solution for the system is cos(4t)u(t). determine, in its simplest form, the forcing function, f(t), applied to the system.

in this case, u(t) is the unit step function and just assume we are working with t>0, thus u(t) = 1 and thus can be ignored. I know how to solve differential equations but never seen this before where u haf to work backward. if anyone could give me any input for how to start it, much appreciated.

Last edited: Sep 18, 2009
2. ### t_n_k AAC Fanatic!

Mar 6, 2009
5,448
790
I get f(t)=1.25cos(4t-36.87°)

Not sure if I'm on the right track ......

3. ### squirby Thread Starter Member

Aug 21, 2009
15
0
that isn't the answer i got here of 5cos(4t + arctan(4/3)) which is the given answer. however, i do not know how they got this nor how to start.

4. ### squirby Thread Starter Member

Aug 21, 2009
15
0
wonderin, anyone still haf ideas? like any steps they can share to start this question off? i'm really desperate been googlin for an example question but can't find it. i just need someone to give me steps to follow... thx

5. ### t_n_k AAC Fanatic!

Mar 6, 2009
5,448
790
Had another look at the problem - I've decided my answer is wrong. I'm not sure whether you have supplied all the information needed for a solution.

But consider this ...

Using the answer you gave and the stated transient + steady state output response, can you derive the system transfer function? If so, does that transfer function have a physical equivalent - such as a real (passive?) circuit or network.

Did the question come from class notes or a text book?

6. ### t_n_k AAC Fanatic!

Mar 6, 2009
5,448
790
Are you familiar with Laplace Transforms?
I might be able to engage in a meaningful (but not necessarily successful!) discussion about the problem if you have used Laplace.
Otherwise hopefully someone else has some suggestions.

7. ### squirby Thread Starter Member

Aug 21, 2009
15
0
thanks for takin the time to reply t n k. i figured the answer out eventually.
first you note that as you are given only 1 complementary solution, it is a first order equation. secondly, you note that the root of the first order eqn is 3 (determined from the complementary solution).

thus you haf an equation of the form dy/dt + 3 y = f(t). as you are given a particular soln, you substitute that into the above equation to find f(t). once you find that, you transform that using a method i mentioned in another of my posts.

anyway, if anyone needs further clarification, feel free to ask.

8. ### t_n_k AAC Fanatic!

Mar 6, 2009
5,448
790
Hi squirby,

Ok well done!

Are you able to explain how you would obtain the given complementary solution
8exp(-3t) for the derived forcing function f(t)=5cos(4t+arctan(4/3))?

Rgds,

t_n_k

9. ### t_n_k AAC Fanatic!

Mar 6, 2009
5,448
790
The reason I ask this is that after some thought it seems to me there is a problem with the solution.

I'll solve it using Laplace for reasons of simplicity.

If indeed the differential equation describes a first order response (lagging) then the system producing that response will have the form

G(s)=1/(s+3)

If f(t)=5cos(4t+θ) where θ=arctan(4/3)

then the equivalent Laplace form of f(t) is

F(s) = 5(s.cosθ-4.sinθ)/(s^2+16) = (3s-16)/(s^2+16)

The general response in the Laplace domain is then

Y(s)=G(s)*F(s) = (3s-16)/[(s+3)(S^2+16)]

which after some manipulation reduces to

Y(s) = s/(s^2+16) - 1/(s+3)

And the time domain equivalent response then falls out as

y(t) = cos(4t)-exp(-3t)

Not

y(t) = cos(4t)-8exp(-3t) which is what your original post in effect states.

So I believe your solution is incorrect.