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1. Knight P11.74 Given the equation of position versus time, x(t) = Ae-t cos(wt + o) Show that the time derivatives of this equation are: dx dt d²x dt2 - Aße-t cos(wt + o) - Awe-t sin(wt + o) =(A3² - Aw²) e-t cos(wt + 0) +2Aßwe-t sin(wt + o)

1. Knight P11.74 Given the equation of position versus time, x(t) = Ae-t cos(wt + o) Show that the time derivatives of this equation are: dx dt d²x dt2 - Aße-t cos(wt + o) - Awe-t sin(wt + o) =(A3² - Aw²) e-t cos(wt + 0) +2Aßwe-t sin(wt + o)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Questions In the following three exercises you will solve the Newton's second law equation for a horizontal oscillating spring with spring constant, k, and linear drag coefficient, b. 1. Knight P11.74 Given the equation of position versus time, 2 x(t) = Ae-t cos(wt + o) Show that the time derivatives of this equation are: dx Aße-t cos(wt + o) dt == d²x dt² - Awe-Bt =(A3² - Aw²) e-Bt cos(wt + po) +2Aßwe-t sin(wt + o) t sin(wt + o) Substitute your answer from Q1 into the
Transcribed Image Text:Questions In the following three exercises you will solve the Newton's second law equation for a horizontal oscillating spring with spring constant, k, and linear drag coefficient, b. 1. Knight P11.74 Given the equation of position versus time, 2 x(t) = Ae-t cos(wt + o) Show that the time derivatives of this equation are: dx Aße-t cos(wt + o) dt == d²x dt² - Awe-Bt =(A3² - Aw²) e-Bt cos(wt + po) +2Aßwe-t sin(wt + o) t sin(wt + o) Substitute your answer from Q1 into the
Useful information Product rule Trig derivatives d(uv) = du v + u dv d du d du cos au = -a sin au sin au= a cos au Derivative of natural log base d - eau = aeau du Sine plus cosine equation: if the sum of sines and co- is zero for all angles, then the coefficients of the sine and cosines are themselves identically zero 0= P cos u + Q sin u P = 0 Q=0
Transcribed Image Text:Useful information Product rule Trig derivatives d(uv) = du v + u dv d du d du cos au = -a sin au sin au= a cos au Derivative of natural log base d - eau = aeau du Sine plus cosine equation: if the sum of sines and co- is zero for all angles, then the coefficients of the sine and cosines are themselves identically zero 0= P cos u + Q sin u P = 0 Q=0
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