If we have a linear system with the following differential equation: d2y/dt2+7*dy/dt+12*y=dx/dt+5*x So H(s) = (s+5)/((s+4)*(s+3)) = -1/(s+4)+2/(s+3)--> h(t) = (-e^(-4*t)+2*e^(-3*t))*u(t) (Please calculate the unit impulse response by Laplace transform and verify the result by yourself) Why this answer does not satisfy the differential equation? The right answer is: h(t) = (-2*e^(-4*t)+3*e^(-3*t))*u(t)
please look at the following equation and Laplace transform gives the correct result this time: d2y/dt2+3*dy/dt+2*y(t) = dx/dt+5* x(t)--> H(s) = (s+5) / [(s+2)*(s+1)]--> h(t) = ( -3*e^(-2*t)+ 4*e^(-t) ) * u(t)
I think your results are all correct. And in the first post, the "right answer" seems to be incorrect. I cannot find anything wrong with your approach or your results. I re-calculated the results using the method of residues and got the same results. Cheers, Vahe
The right answer satisfies the differential equation. Please give that a try. It is strange why the impulse response does not satisfy the differential equation while it is its answer by Laplace transform.
It's not clear why you say the method gives the wrong answer because it seems to satisfy the diff equation.
If h(t)=(-e^(-4*t)+2*e^(-3*t))*u(t) is the impulse response, as its Laplace transform suggests, so it must satisfy "d2y/dt2+7*dy/dt+12*y=dx/dt+5*x" so(using the chain rule and our input is the unit impulse function, delta(t) ): y(t) = (-e^(-4*t)+2*e^(-3*t))*u(t) and x(t) = delta(t) dy/dt = (4*e^(-4*t)-6*e^(-3*t))*u(t)+(-e^(-4*t)+2*e^(-3*t)) * delta(t) d2y/dt2 = (-16*e^(-4*t)+18*e^(-3*t))*u(t)+[4*e^(-4*t)-6*e^(-3*t)+4*e^(-4*t)-6*e^(-3*t)]*delta(t)+ (-e^(-4*t)+2*e^(-3*t))*d(delta(t))/dt d2y/dt2+7*dy/dt+12*y = (-e^(-4*t)+2*e^(-3*t))*d(delta(t))/dt + [8*e^(-4*t)-12*e^(-3*t)-7*e^(-4*t)+14*e^(-3*t)]*delta(t)+ [-16*e^(-4*t)+18*e^(-3*t)+28*e^(-4*t)-42*e^(-3*t)-12*e^(-4*t)+24*e^(-3*t))*u(t)= (4*e^(-4*t)-6*e^(-3*t)) d(delta(t))/dt + [e^(-4*t)+2*e^(-3*t)]*delta(t)+0*u(t) ---> d2y/dt2+7*dy/dt+12*y = (-e^(-4*t)+2*e^(-3*t)) *d(delta(t))/dt + [e^(-4*t)+2*e^(-3*t)]*delta(t) = d(delta(t))/dt+5*delta(t) so the coefficient of delta(t) is 3 in the left side while it is 5 in the other side! and it seems the equality can not be satisfied. If (-e^(-4*t)+2*e^(-3*t))*u(t) is the impulse response why it does not satisfy the differential equation?
Ah, I see. You are worried about the point t=0. The equation is OK everywhere but there. I'm not a mathemetician, but I would not expect to have things well behaved at this singularity. The point t=0 becomes a nasty place for differential equations when an input impulse function (and in this case even its derivative) enter. The system impulse response helps you reconstruct the response to other less-nasty input functions, which would then have agreement at t=0. Maybe there is a mathematican, or signals and systems expert, that can shed better light on this, but basically I wouldn't worry about singular points if you are just interested in engineering application of the theory. From a practical point of view, the impulse response is nearly the response you would get from a very narrow input pulse with area of one. In that sense, it satisfies the system differential equation.