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Unformatted text preview: ______________________________________________________________________ 3. (15 pts.) Find a particular integral, y p , of the differential equation 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 anymost n roots....
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 Fall '08
 STAFF
 Vector Space

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