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HW8_solution

# HW8_solution - EEL3105 Fall2011 Homework8 1 d2y dy 6 16 y...

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EEL 3105 Fall 2011 Homework 8 1. Solve the following problems for both of these the differential equations 2 2 2 2 6 16 ( ) 12 16 ( ) d y dy y u t dt dt d y dy y u t dt dt To get ready to solve the problems below, let us first calculate the roots of the corresponding polynomials: 2 2 6 16 12 16. and The roots are 3 7 j   [labeled as 1 2 ,   ] and 10.47, 1.53 [labeled as 3 4 , ] respectively. Thus, the solution the homogeneous part of the first differential equation [I will label it (i)] has the form: ( 3 2.65) ( 3 2.65) 3 [ cos(2.65 ) sin(2.65 )] j t j t t Ae Be or e M t N t     where A,B or M,N are unknown constants that depend on the initial conditions. Note that this function is a decaying sinusoid. Please plot it using MATLAB to understand this important point. Similarly, the solution to the homogeneous part of the second differential equation [labeled as (ii)] has the form 10.47 1.53 t t Ae Be This function, by contrast, has no oscillations but simply decays exponentially to zero. The slower time constant 1.53 determines, to a dominant extent, the rate of decay to zero. Again, please plot using MATLAB to get a sense of this function. We are now ready to attack the problems below. A. Suppose y(0)=0, (0) 0 dy dt and u=U(t) where U(t) is the unit step function. Find y(t).

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