## Transcribed Text

1 1
(6)
1. Consider the matrix A =
0 1 0
.
0 -3 4
Find out whether A is diagonalizable and if so, find a diagonalization of A.
2
1
1
(7)
2. Consider the vectors =
and the vector x=
1
3
2
2
3
Consider the linear subspace W = Span{a1,a2} of R4.
(a) Determine an orthogonal basis of W.
(b) Determine w € W and wit E w with x = w + wR.
2 5
(7)
3. Consider the real system X' = AX, with A =
2 0
and x(t) = [nd
(a) Find the general real solution of this system.
(b) Make a sketch of the phase portrait of this system. If appropriate, explain with a
direction-arrow whether the motion is clockwise or counter clockwise.
(c)
Classify the critical point (0,0) as to type and determine whether it is stable, asymp-
totic stable or unstable.
(15)
4. Consider the heat conduction problem for the function 21(I.,
= un, with 0 < x < 2 en t > 0.
We also have the following initial values and boundary values:
(0, = 1. u(2,t) = 3,
(a) Find the steady-state solution v(I) of the partial differential equation with the given
boundary values.
(b)
Put y(z,t) = 21(I.t) - v(z). Write a partial differential equation for y(x,1 t). together
with initial values and boundary values.
(c) Use the method of separation of variables to solve y(x,t).
(d) Solve u(x,t)

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