Hello All,
I am struggling to find the correct unit impulse response, h(t) for a specific differential equation. I have outlined my methods below and attached my handwritten work:
Differential equation: y''(t)+2*y'(t)+2*y(t)=f'(t)
1) Find the roots of the equations
2) Based on the roots, find the characteristic equation (2 constants: C and theta, this is shown in the attachment)
3) The equation is second order to so let h(0) = 0 and h'(0) = 1 to represent the input being concentrated at 0 for a unit impulse input
4) Use both equations to solve for C and theta
5) Obtain final equation for unit impulse response in terms of time.
My answer is e^t * sin(t)*u(t) where u(t)=unit step when I use convolution. When I use laplace transform to find H(s)=Y(s)/X(s) (Output/Input) and use inverse laplace transform, I obtain sqrt(2)*e^t*cos(t+pi/4). The second answer is correct, but I'm not sure why convolution is giving me a different answer. Could someone please help me locate my error?
I've added the typed version of my work below:
s^2 + 2*s+2 = 0
I found the root using the quadratic formula: s1,2 = 1 +/ j
so the form of the impulse response is C*e^t*cos(t+O) where O=theta
h(t)=C*e^t*cos(t+O)
h'(t)=C*(e^t*cos(t+O)e^t*sin(t+O))
In order to account for the impulse input response let h(0)=0 and h'(0)=1
When I do that:
h(0) = C*e^0*cos(0+O)=C*cos(O)=0 O=theta
h'(0) = C*(e^0*cos(O)e^0*sin(O) )=C*(cos(O)+sin(O))=1
I used elimination to remove the C*cos(O) term:
C*cos(O)=0
C*cos(O)C*sin(O)=1
Result: C*sin(O)=1
Then I squared each side of the equations:
1) (C*cos(O))^2=(0)^2>(C^2)*cos(O)^2=0
2) (C*sin(O))^2=1^2> (C^2)*sin(O)^2=1
When I add (1) and (2), my result becomes:
C^2*(cos(O)^2+sin(O)^2)=1, using the identity sin(O)^2 + cos(O)^2 = 1,
I am left with: C^2=1 so C=1 (it was sqrt(2) when I used laplace transform referring to Y(s)/X(s) where Y(s)=output and X(s)=input)
To find O(theta) I substituted C into C*sin(O)=1 where
sin(O)=1. So O=pi/2 to cancel out the negative sign.
My final result is C*e^t*cos(t+O)>e^t*cos(tpi/2)
When I redid the problem with laplace transform , I obtained sqrt(2)*e^t*cos(t+pi/4).
I am struggling to find the correct unit impulse response, h(t) for a specific differential equation. I have outlined my methods below and attached my handwritten work:
Differential equation: y''(t)+2*y'(t)+2*y(t)=f'(t)
1) Find the roots of the equations
2) Based on the roots, find the characteristic equation (2 constants: C and theta, this is shown in the attachment)
3) The equation is second order to so let h(0) = 0 and h'(0) = 1 to represent the input being concentrated at 0 for a unit impulse input
4) Use both equations to solve for C and theta
5) Obtain final equation for unit impulse response in terms of time.
My answer is e^t * sin(t)*u(t) where u(t)=unit step when I use convolution. When I use laplace transform to find H(s)=Y(s)/X(s) (Output/Input) and use inverse laplace transform, I obtain sqrt(2)*e^t*cos(t+pi/4). The second answer is correct, but I'm not sure why convolution is giving me a different answer. Could someone please help me locate my error?
I've added the typed version of my work below:
s^2 + 2*s+2 = 0
I found the root using the quadratic formula: s1,2 = 1 +/ j
so the form of the impulse response is C*e^t*cos(t+O) where O=theta
h(t)=C*e^t*cos(t+O)
h'(t)=C*(e^t*cos(t+O)e^t*sin(t+O))
In order to account for the impulse input response let h(0)=0 and h'(0)=1
When I do that:
h(0) = C*e^0*cos(0+O)=C*cos(O)=0 O=theta
h'(0) = C*(e^0*cos(O)e^0*sin(O) )=C*(cos(O)+sin(O))=1
I used elimination to remove the C*cos(O) term:
C*cos(O)=0
C*cos(O)C*sin(O)=1
Result: C*sin(O)=1
Then I squared each side of the equations:
1) (C*cos(O))^2=(0)^2>(C^2)*cos(O)^2=0
2) (C*sin(O))^2=1^2> (C^2)*sin(O)^2=1
When I add (1) and (2), my result becomes:
C^2*(cos(O)^2+sin(O)^2)=1, using the identity sin(O)^2 + cos(O)^2 = 1,
I am left with: C^2=1 so C=1 (it was sqrt(2) when I used laplace transform referring to Y(s)/X(s) where Y(s)=output and X(s)=input)
To find O(theta) I substituted C into C*sin(O)=1 where
sin(O)=1. So O=pi/2 to cancel out the negative sign.
My final result is C*e^t*cos(t+O)>e^t*cos(tpi/2)
When I redid the problem with laplace transform , I obtained sqrt(2)*e^t*cos(t+pi/4).
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