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Consider the harmonic oscillator equation (with m = 1) x" + bx' + kx = 0 where b > 0 and k > 0. Identify the regions in the relevant portion of the bk-plane where the corresponding system has similar phase portraits.

Consider the harmonic oscillator equation (with m = 1) x" + bx' + kx = 0 where b > 0 and k > 0. Identify the regions in the relevant portion of the bk-plane where the corresponding system has similar phase portraits.

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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3. Consider the harmonic oscillator equation (with m = 1) x" + bx' + kx = 0 where b > 0 and k > 0. Identify the regions in the relevant portion of the bk-plane where the corresponding system has similar phase portraits.
Transcribed Image Text:3. Consider the harmonic oscillator equation (with m = 1) x" + bx' + kx = 0 where b > 0 and k > 0. Identify the regions in the relevant portion of the bk-plane where the corresponding system has similar phase portraits.
Primero, por lo visto en las primeras páginas del capitulo 2, (*) se puede tronsformar en un sistema plono de la siquiente manera: x'= y'= -kx- by A hora, lo podemos expresor de la siguiente monera 1 X'= AX, donde A = -k -b TrA = -b y DetA = K, vemos que la troza es siempre menor o igual que cero. k b²:4K Estamos interesados solo en la región b20 y k>o que es cuadronte . porte del primer cuadrante. Con oyuda del plono T-0. ubicamoS O< DetA =K. • 0=b= -b= Tr A Tenemos centros Como b>o ) -b=TrA<O A= b² -4K <o Tenemos espirales estables A= b2 -4k =0 y TrA co, entonces los Ya que valores propios son iguales y negatNos. Tenemos nodos impropios estables. • -b = TrA<O Y A=b²-4K >0, entonces
Transcribed Image Text:Primero, por lo visto en las primeras páginas del capitulo 2, (*) se puede tronsformar en un sistema plono de la siquiente manera: x'= y'= -kx- by A hora, lo podemos expresor de la siguiente monera 1 X'= AX, donde A = -k -b TrA = -b y DetA = K, vemos que la troza es siempre menor o igual que cero. k b²:4K Estamos interesados solo en la región b20 y k>o que es cuadronte . porte del primer cuadrante. Con oyuda del plono T-0. ubicamoS O< DetA =K. • 0=b= -b= Tr A Tenemos centros Como b>o ) -b=TrA<O A= b² -4K <o Tenemos espirales estables A= b2 -4k =0 y TrA co, entonces los Ya que valores propios son iguales y negatNos. Tenemos nodos impropios estables. • -b = TrA<O Y A=b²-4K >0, entonces
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