HW6_soln

HW6_soln - fhb 14 Section 4.7 Problem 18 We started with the differential equation ݔ ᇱᇱ ൅ 2ݔ ᇱ ൅ 2ݔ ൌ cos ሺ2ݐ where the right

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Unformatted text preview: fhb 14 Section 4.7 Problem 18 We started with the differential equation: ݔ ᇱᇱ ൅ 2ݔ ᇱ ൅ 2ݔ ൌ cos ሺ2ݐሻ where the right hand side ܨሺݐሻ ൌ cos ሺ2ݐሻ is the external force acting on the system. From the analysis, we saw that the general solution is: ݔሺݐሻ ൌ ଵ ଵ଴ ݁ ି௧ cosሺݐሻ ൅ ଵ଻ ଵ଴ ݁ ି௧ sinሺݐሻ െ ଵ ଵ଴ cosሺ2ݐሻ ൅ ଶ ହ sinሺ2ݐሻ Any terms that contain an exponential to a negative power (i.e. ݁ ି௧ terms in the above equation) will go to zero as we increase ݐ . These terms form the transient component of the general solution; they are the part of the solution that becomes negligible as ݐ increases. The following is an example of the transient component versus the steady ‐ state component of a general solution. Say you excite an electric circuit with a time ‐ varying voltage; there will be a large initial change in voltage (and current) throughout the circuit. As time goes on, the circuit gets used this input voltagein voltage (and current) throughout the circuit....
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This note was uploaded on 04/02/2008 for the course MATH 33B taught by Professor Staff during the Spring '07 term at UCLA.

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HW6_soln - fhb 14 Section 4.7 Problem 18 We started with the differential equation ݔ ᇱᇱ ൅ 2ݔ ᇱ ൅ 2ݔ ൌ cos ሺ2ݐ where the right

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