Complex Numbers
Convert this to polar form:
z = 2 + j3
*argument θ in radians
z = √13[cos(0.9828) + jsin(0.9828)]
z = √13cis0.9828
z = √13ej0.9828
z = √13∠0.9828
What is the auxilliary equation of this homogeneous differential equation?
(D3 - 6D2 + 12D - 8)y = 0
m3 - 6m2 + 12m - 8 = 0
What is the Laplace Transform of f(t) = t2 + 3t - 2
F(s) = 2/s3 + 3/s2 - 2s
Evaluate: ln(30 - j3)
* angle θ in radians
z = 3.406 - j0.997 (not adjusted angle)
z = 3.406 + j5.28 (adjusted angle)
What is the complementary solution of this differential equation?
y'' - 4y' - 5y = 0
yc = C1e5x + C2e-x
What is the Laplace Transform of f(t) = te-2t
F(s) = 1/(s+2)2
Let z = 8 + j5 and w = 2 – j
Perform this operation: z` + zw
29 - j3
Compute the Wronskian value of this differential equation:
y'' + y = tan x
W(y1, y2) = 1
W(y1, y2) = y1*y2' - y1'*y2
Find the Inverse Laplace transform of
F(s) = s+5/s2+4
f(t) = cos2t + 5/2 sin2t
Apply De Moivre's theorem
z = (16ejπ/4)6
z = 16,777,216ej3π/2
Guess the particular solution of this non-homogeneous differential equation based on its Yc:
y'' - 2y' + y = ex
Yc = (C1 + C2x)ex
Yp = ?
yp = Ax2ex
Write the Partially Decomposed Fraction terms of this s-domain function:
F(s) = s2 + 7s + 2 / (s3 - 1)(s2 + 1)
= [(Ax2 + Bx + C) / (s3 - 1)] + [(Dx + E) / (s2 + 1)]
Determine all the 2nd roots of
z = 1 – j3
*argument θ in radians
final answers in standard form
(adjusted)
w0 = -1.438 + j1.045
w1 = 1.438 - j1.045
(non-adjusted)
w0 = 1.442 - j1.040
w1 = -1.442 + j1.040
When finding for a particular solution using Variation of Parameter, you need to impose this constraint auxiliary equation to simplify computations and to eliminate higher derivatives.
What is the imposed constraint auxiliary equation?
u1'y1 + u2'y2 = 0
Find the Laplace transform of f(t) = t2sint
using derivative of transform property:
L{tnf(t)} = (-1)n dn/dsn F(s)
F(s) = 2(3s2 - 1) / (s2 + 1)3