Inter American University of Puerto Rico
Bayamón Campus
School of Engineering
Department of Electrical Engineering
ELEN 3301 – Electric Circuits I
Problem Set 6
Due Wednesday, October 13
Problem 1: Show that
v(t) = Ke−αt sin(ωd t + φ)
is the general solution of the differential equation
d2 v
dv
+ 2α + ω02 v = 0 where ω0 > α
2
dt
dt
p
if ωd = ω02 − α2 . The method will be outlined in the following parts.
(A) Show that
dv
= −αKe−αt sin(ωd t + φ) + ωd Ke−αt cos(ωd t + φ)
dt
(B) Show that
d2 v
= α2 Ke−αt sin(ωd t + φ) − 2αωd Ke−αt cos(ωd t + φ) − ωd2 Ke−αt sin(ωd t + φ)
dt2
2
(C) Substitute the expressions for ddt2v , dv
and v into the differential equation. Group the
dt
coefficients of the sine and cosine terms and equate each group to zero. p
You should be
left with two algebraic equations. Show that the equations hold if ωd = ω02 − α2 .
(D) You have found the general solution for the differential equation given in the problem
statement. Given this solution, find the solution for the differential equation
d2 v
+ ω02 v = 0
dt2
(E) Show that the solution for the differential equation found in the previous part can be
converted to the form
v(t) = A sin(ω0 t) + B cos(ω0 t)
Hint: sin(x + y) = sin x cos y + cos x sin y.