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[tex]y'' + 6y' + 4y = x(t)[/tex]
I found the eigenvalues to be -.764 and -5.24, no problems here. Next, since the order of the response is greater than the order of the excitation, I assumed the unit response took the form:
[tex] h(t) = k_1e^{-.764t} + k_2e^{-5.24t}[/tex]
So now I must find the values for k_1 and k_2. If I integrate both sides from -0 to +0, I should find one equation that will help in this endeavor. This is where I run into some problems.
[tex]\int\limits_{-0}^{+0}h''(t)\, dx + 6\int\limits_{-0}^{+0}h'(t)\, dx + 4\int\limits_{-0}^{+0}h(t)\, dx = \int\limits_{-0}^{+0}\delta(t)\, dx[/tex]
which becomes:
[tex] h'(+0) - h'(-0) + 6[h(+0) - h(-0)] + 4\int\limits_{-0}^{+0}h(t)\, dx = u(+0) - u(-0)[/tex]
Now, h(t) has no impulse in it. Therefore, an integral from -0 to +0 of h(t) = 0. Also, since the only excitation is the impulse, occurring at t = 0, and this system is causal, all h(t) and its derivatives evaluated at -0 evaluate to zero. Further, a step function at -0 is zero and at +0 is 1:
[tex] h'(+0) + 6h(+0) = 1[/tex]
I then substitute in h'(+0) and 6h(+0):
[tex] h'(+0) + 6h(+0) = -.764K_1 - 5.24K_2 + 6k_1 + 6K_2 = 1[/tex]
The problem is that the first equation generated in the solutions does not have the additional 6k_1 and 6K_2. Can someone explain to me why [tex]6\int\limits_{-0}^{+0}h'(t)\, dx[/tex] evaluates to zero?
I've completed the problem, and some how got the same(right) answer? How can this be? When I solved these two systems of equations (the first, what I derived and the second what the solutions manual derived) i got the same answer.
[tex]5.236k_1 + .760k_2 = 1[/tex]
[tex]k_1 + k_2 = 0[/tex]
theirs:
[tex] -.760k_1 - 5.24k_2 = 1[/tex]
[tex] k_1 + k_2 = 0[/tex]
I found the eigenvalues to be -.764 and -5.24, no problems here. Next, since the order of the response is greater than the order of the excitation, I assumed the unit response took the form:
[tex] h(t) = k_1e^{-.764t} + k_2e^{-5.24t}[/tex]
So now I must find the values for k_1 and k_2. If I integrate both sides from -0 to +0, I should find one equation that will help in this endeavor. This is where I run into some problems.
[tex]\int\limits_{-0}^{+0}h''(t)\, dx + 6\int\limits_{-0}^{+0}h'(t)\, dx + 4\int\limits_{-0}^{+0}h(t)\, dx = \int\limits_{-0}^{+0}\delta(t)\, dx[/tex]
which becomes:
[tex] h'(+0) - h'(-0) + 6[h(+0) - h(-0)] + 4\int\limits_{-0}^{+0}h(t)\, dx = u(+0) - u(-0)[/tex]
Now, h(t) has no impulse in it. Therefore, an integral from -0 to +0 of h(t) = 0. Also, since the only excitation is the impulse, occurring at t = 0, and this system is causal, all h(t) and its derivatives evaluated at -0 evaluate to zero. Further, a step function at -0 is zero and at +0 is 1:
[tex] h'(+0) + 6h(+0) = 1[/tex]
I then substitute in h'(+0) and 6h(+0):
[tex] h'(+0) + 6h(+0) = -.764K_1 - 5.24K_2 + 6k_1 + 6K_2 = 1[/tex]
The problem is that the first equation generated in the solutions does not have the additional 6k_1 and 6K_2. Can someone explain to me why [tex]6\int\limits_{-0}^{+0}h'(t)\, dx[/tex] evaluates to zero?
I've completed the problem, and some how got the same(right) answer? How can this be? When I solved these two systems of equations (the first, what I derived and the second what the solutions manual derived) i got the same answer.
[tex]5.236k_1 + .760k_2 = 1[/tex]
[tex]k_1 + k_2 = 0[/tex]
theirs:
[tex] -.760k_1 - 5.24k_2 = 1[/tex]
[tex] k_1 + k_2 = 0[/tex]
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