Laplace Transform-Differenrial Equations-Exam Papers, Exams of Differential Equations

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1. (a) (15 points) If f �^ ∈ E and f is continuous, show that lims→∞ sF (s) = f (0).

(b) (5 points) Can F (s) = 1 be the Laplace transform of a function f ∈ E?

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2. (a) (10 points) Show that the solution of the initial value problem

y��^ + 2y�^ + 2y = f (t), y(0) = y�(0) = 0

is (^) � (^) t

y(t) = e−(t−t^1 )f (t 1 ) sin(t − t 1 )dt 1. 0

(b) (10 points) Show that if f (t) = δ(t − π) then the solution of the initial value problem in part (a) is y(t) = h(t − π)e−(t−π)^ sin(t − π).

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4. Let A =

(a) (10 points) Find eigenvalues and eigenvectors of A.

(b) (10 points) Find the general solution of x �^ x 2 y =^ A^ y +^1 e

−t (^).

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5. Let A =

(a) (8 points) Find eigenvalues and eigenvectors of A.

(b) (7 points) Find the solution of the initial value problem

x � x x(0) 3 y =^ A^ y ,^ y(0) =^2.

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