Differential and Difference Equations
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Part a. Solve the following differential equation for t\ge0 assuming the initial conditions y(0) = 1 and \left.{dy(t)\over dt}\right|_{t=0}=2.
y(t)+3{dy(t)\over dt}+2{d^2y(t)\over dt^2}=1
Express the resulting y(t) in closed form, with no integrals or derivatives.
[Hint: assume the homogeneous solution has the form Ae^{s_1t}+Be^{s_2t}.]
Part b. Solve the following difference equation for n\ge0 assuming the initial conditions y[0] = 1 and y[-1] = 2.
8y[n]-6y[n-1]+y[n-2]=1
Express the resulting y[n] in closed form (with no infinite sums).
[Hint: assume the homogeneous solution has the form Az_1^n+Bz_2^n.]
Part c. Assume that a continuous time system can be described by a linear differential equation with constant coefficients. If the input x(t) is equal to 1 for t\gt0, then the output y(t) is given by
y(t)={1\over6}e^{-t/4}+{1\over2}e^{-t/2}+{1\over3}
for t\gt0.
Determine a differential equation of the following form:
y(t)+A{dy(t)\over dt}+B{d^2y(t)\over dt^2}=Cx(t)
to describe the relation between x(t) and y(t).
Upload your solution to this problem (including parts a, b, and c) as a single pdf file.
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