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MAT485-2006Spring

# MAT485-2006Spring - M AT 4 85 S pring 2 006 N AME F inal E...

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MAT 485, Spring 2006 NAME: Final Exam 1

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2 1) (18 pts) I)Let A = ( 3 20 2) -1 ' B = (2 3 -2) 4 ' C = (1 2 -3 -4 5) 6 Find A + B, 2B, AB, and CTB.
3 2) (18 pts) Solve by the Gauss-Jordan procedure: x + 3y + 2z = 5 2x + 7y + 4z = 8 2x + 8y + 3z = 2.

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4 3) (18 pts) A transformation T(v) = Av is given by the matrix 1 -4 -3 -7) A = 2 -1 1 7 . ( 1 2 3 11 Find the kernel, nullity and rank of T and the general solution of the homogeneous system Ax=O.
5 4) (18 pts) Solve the initial value problem: , y = ycotx, y( 1r /2) = 1r /2.

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6 5) (18 pts) In RL-circuit the current I satisfies the equation Find the current at t = 2 if the resistance R is t ohms, the inductance L is 1 henry, E(t) = t volt, and the initial current is 2.
7 6) (18 pts) Find the general solution of the differential equation: y" - 3y' + 2y = t 2 .

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8 7) (18 pts) Use the method of variation of parameters to find the general solution of the differential equations: y" - 4y' + 5y = e 2x tan x.
9 8) (18 pts) Use the Laplace transform to solve the initial value problem: y' - y = 2step(t - 2), y(O) = 1.

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10 9) (18 pts) Solve the initial value problem: (3 4) (i). x I = 3 2 x, x(O) =
11 10) (20 pts) Solve the system

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12 11) (18 pts) Find all equilibrium points of the given systems and classify their stability and geometry: x' =x 2 + Y y' =x _ y2
Table of Laplace Transforms Clf(t)) = F(s) ==

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