Differential and Difference Equations

The questions below are due on Thursday September 17, 2026; 02:00:00 PM.

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This problem illustrates how to construct solutions to linear differential and difference equations with constant coefficients using just simple rules of algebra and differentiation. Advanced methods of solving differential equations (as contained in 18.03) or difference equations (as contained in 18.06) are not required. If you have difficulties with any of the following problems, please discuss with a staff member at office hours.

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.]

Upload your solution to this problem (including parts a and b) as a single pdf file.
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