nagle_differential_equations_ISM_Part53

nagle_differential_equations_ISM_Part53 - Chapter 7 Using...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Chapter 7 Using linearity of the inverse Laplace transform and formula (6) in Section 7.6, we get x ( t ) =- 1 2 + 2 3 e- t- 1 6 e 2 t- 1 4- x 2- 1 3 e- x + 1 12 e 2 x x = t- 3 u ( t- 3) =- 1 2 + 2 e- t 3- e 2 t 6- 1 4- t- 3 2- e 3- t 3 + e 2 t- 6 12 u ( t- 3) . Since y = x- x (see the first equation in the given system), we obtain y ( t ) =- 1 2 + 4 e- t 3 + e 2 t 6- 3 4- t- 3 2- 2 e 3- t 3- e 2 t- 6 12 u ( t- 3) . 14. Since L{ x 00 } ( s ) = s 2 X ( s )- sx (0)- x (0) = s 2 X ( s )- s, L{ y 00 } ( s ) = s 2 Y ( s )- sy (0)- y (0) = s 2 Y ( s ) , applying the Laplace transform to the given equations yields s 2 X ( s )- s = Y ( s ) + e- s /s s 2 Y ( s ) = X ( s ) + (1 /s )- e- s /s s 2 X ( s )- Y ( s ) = s + ( e- s /s )- X ( s ) + s 2 Y ( s ) = (1 /s )- ( e- 3 s /s ) . Solving for X ( s ) yields X ( s ) = s 4 + 1 s ( s 4- 1) + s 2- 1 s ( s 4- 1) e- s = s 4 + 1 s ( s 4- 1) + 1 s ( s 2 + 1) e- s =- 1 s + 1 2 1 s + 1 + 1 2 1 s- 1 + s s 2 + 1 + 1 s- s s 2 + 1 e- s . Using linearity of the inverse Laplace transform and formula (6) in Section 7.6, we get x ( t ) =- 1 + e- t 2 + e t 2 + cos t + [1- cos( t- 1)] u ( t- 1) = cosh t + cos t- 1 + [1- cos( t- 1)] u ( t- 1) . Since y = x 00- u ( t- 1) (see the first equation in the system), after some algebra we obtain y ( t ) = cosh t- cos t- [1- cos( t- 1)] u ( t- 1) . 16. First, note that the initial conditions are given at the point t = . Thus, for the Laplace transform method, we have to shift the argument to get zero initial point. Let us denote w ( t ) := x ( t + 1) and v ( t ) := y ( t + ) . 256 Exercises 7.9 The chain rule yields w ( t ) = x ( t + )( t + ) = x ( t + ) , v ( t ) = y ( t + )( t + ) = y ( t + ) ....
View Full Document

Page1 / 5

nagle_differential_equations_ISM_Part53 - Chapter 7 Using...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online