x^s(A0 + A1x + A2x + ... + Anx^n) Cosine Addition/Subraction Trig ID Inverse of a 2x2 matrix Pm(x) = b0 + b1x + b2x^2 + ... + bnx^n period

Pm(x) = b0 + b1x + b2x^2 + ... + bnx^n

x(t) = Acos(w0t) + Bsin(w0t) = Ccos(w0 - a) Conditions c = 0 F(t) = 0 Solution of an undamped and forced system mx'' + cx' + kx = Mcos(wt) Pm(x) = b0 + b1x + b2x^2 + ... + bnx^n Solution of an undamped and free system mx'' + cx' + kx = F(t) Complex Eigenvalue Graphs

Solution of an undamped and free system mx'' + cx' + kx = F(t)

MA266 Midterm 2

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Calculus - Geometry

user_hodr Created by 6 mon ago

Cards in this deck(17)

x^s(A0 + A1x + A2x + ... + Anx^n)

x^s(Acoskx + Bsinkx)

x^s(Acoskx + Bsinkx)e^rx

x^s(A0 + A1x + A2x + ... + Anx^n)e^rx

x^s(A0 + A1x + A2x + ... + Anx^n)cosKx + x^s(B0 + B1x + B2x + ... + Bnx^n)sinkx

in ay'' +by' +cy = f(x), y1 and y2 are solutions W(y1, y2) = [y1(x), y2(x)] [y1'(x), y2'(x)] yp(x) = −y1(x)[∫y2(x)f(x)dx / W (x)] + y2(x)[∫ y1(x)f(x)dx / W (x)]

x(t) = Acos(w0t) + Bsin(w0t) = Ccos(w0 - a) Conditions c = 0 F(t) = 0

C = sqrt(A^2 + B^2)

w0 = sqrt(k / m)

2pi / w

if A > 0 a = arctan(B/A) if A < 0 a = pi + arctan(B/A)

d = a / w0

cos(a + b) = cosacosb - sinasinb cos(a-b) = cosacosb + sinasinb

x(t) = Acos(w0t) + Bsin(w0t) + xp(t) if w = w0 xp(t) = Ctcos(w0t) +Dtsin(w0t) if w (not = to) w0 Ccos(w0t) +Dsin(w0t) if c^2 < 4km so that the roots are complex (a + or - bi) x(t) = e^at(Acos(bt) + Bsin(bt) + Ccos(w0t) + Dsin(w0t) (same as xp(t)) Conditions c = 0 F(t) (not = to) 0

(1/ad-bc) * [d -b] [-c a]

If both lambda have opposite sign: saddle point If both lambda have same sign: node if lambdas are equal: proper if lanbdas are not equal: improper if 0 < lambdas: source if 0 > lambdas: sink If lambda = p + or - iq p > 0: spiral source p = 0: center p < 0: spiral sink

Φ(t) = [V1 V2], where V are 2x1 eigenvectors e^at = Φ(t)Φ^-1(t)

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MA266 Midterm 2

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