MAP
de-t2-a(1)

# Thus we must use variation of parameters to nab the

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Obviously the driving function here is NOT a UC function. Thus, we must use variation of parameters to nab the culprit. Corresponding Homogeneous: F.S. = {sin(2 x ), cos(2 x )}. If y p = v 1 cos(2 x ) + v 2 sin(2 x ), then v 1 and v 2 are solutions to the following system: Solving the system yields v 1 = -2 and v 2 = 2cot(2 x ). Thus, by integrating, we obtain Thus, a particular integral of the ODE above is ______________________________________________________________________ 4. (10 pts.) Set up the correct linear combination of undetermined coefficient functions you would use to find a particular integral, y p , of the O.D.E. First, the corresponding homogeneous equation is which has an auxiliary equation given by 0 = m 2 - 6 m + 10. Thus, m = 3 + i or m = 3 - i , and a fundamental set of solutions for the corresponding homogeneous equation is { exp(3 x )cos( x ), exp(3 x )sin( x ) }. Taking this into account, we may now write ______________________________________________________________________ Bonus Noise: If , then the first n derivatives are nonzero and given by Higher order derivatives are zero. If f is a solution to (*) where m > n , then by direct substitution (**) The left side of (**) is a nonzero polynomial of degree at most n if there is some a k nonzero for some k with 0 k n . Such a polynomial has at most n roots. So in this case, we would not have a function identity on any interval of the line. It follows that a k = 0 for every k with 0 k n . Consequently, the fundamental set of solutions includes the following:

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TEST2/MAP2302 Page 3 of 4 ______________________________________________________________________ 5. (10 pts.) The factored auxiliary equation of a certain homogeneous linear O.D.E. with real constant coefficients is as follows: Note: The plus sign in the factor ( m + ( π - i )) 2
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