Math 2280 - Exam 3University of UtahSpring 2009Name:1Defective Eigenvalues - Solve the system of ODEs:x′=1 0 018 7 4−27 −9 −5x.(10 points).2Continued...3Matrix Exponentials - Calculate eAtfor the matrix:A =3 0 −35 0 73 0 −3.(5 points).4Undetermined Coefficients - Apply the method of undetermined coeffi-cients to find a particular solution for the system of ODEs:x′= x − 5y + 2 sin t,y′= x − y − 3 cos t.(5 points).5Continued...6Laplace Transforms - Calculate the Laplace transform of the function:f(t) = t2directly from the definition of the Laplace transform. (5 points).7Solving ODEs with Laplace Transforms - Use Laplace transform meth-ods to solve the initial value problem:x′′− 6x′+ 8x = 2;x(0) = x′(0) = 0.(10 points)8Continued...9Convolutions and Products - Using the definition of convolution calcu-late the convolution product:f(t) ∗ g(t)where f(t) = t2and g(t) = t. (7 points)Calculate the Laplace transform L(f(t) ∗ g(t)). (3 points)10Delta Functions - Solve the initial value problem:x′′+ 2x′+ x = δ(t) − δ(t − 2);x(0) = x′(0) = 2.(10 points).11Continued...12You may find the following formulas useful:L(f(t)) =Z∞0e−stf(t)dtf(t) ∗ g(t) =Zt0f(τ)g(t − τ)dτL(f(t) ∗ g(t)) = L(f (t)) · L(g(t))L(u(t − a)f(t − a)) = e−asF (s)L(tneat) =n!(s −
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