Class Participation and Summary

Student Name:                                                                  Student ID: 

  • Change the second order equation    y'' +  sin y' +  cos y = 0,  y(0) = -1,  y'(0) = 1  to a first order system.

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  • Given an initial value problem     y' =  f(t, y),   y(0) =  y0,   what is the equivalent integral equation?

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  • What is the local truncation error of the Euler's method?    What is the order of the local truncation error? What is the order of the global error?

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  • Write down the formula of the improved Euler's method and the implicit trapezoidal method.

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  • Given the following initial value problem:
    • y1' =   y1  -  sin y2                            y1(0) = 0
    y2' =  y22 +  ey1 - 1                          y2 (0) = 1
     
  • Write down the equations that determine the equilibriums?

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  • Is (0,0) an equilibrium?

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  • What is the Jacobian matrix of the system at (0, 0)?

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  • Assume the eigenvalues of the Jacobian matrix is 1.5 + 0.86 i and 1.5-0.86 i. How do we call such an equilibrium?

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  • Write one step calculation for the Euler's method and backward Euler's method with  h = 0.1.
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