Midterm exam - revision problems

1. Find the general solution and the particular solution for each of the following linearnon-autonomous di¤erential equations.

(a) tx+ et � t �x = 0; t > 0, x(1) = 0(b)

�x+ x = tet; x(0) = 1

2. Solve the following Bernoulli equation assuming t > 0; y > 0 :�y � 4y = 2etpy

3. For the following di¤erential equation�x = 3

px� x

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(a) Find all steady-state(s) (equilibria, �xed points).

(b) Using linear approximation describe local asymptotic stability of the steady-state(s).

(c) Sketch a phase diagram and establish the stability properties of the steady-state(s).Compare with your answer in (b).

4. Consider the following non-linear systems of ODEs. For each of them draw a phasediagram and establish the stability properties of the steady-state(s), sketch samplephase curves.

(a)

( �x = x2 + y2 � 8�y = x+ y

(b)

( �x = 3x+ y�y = �x2 � y + 4

5. Find extreme points for the functions

(a) f(x; y) = x3 + y3 � 3xy(b) f(x; y) = ex+y + ex�y � 3

2x� 1

2y

(c) f(x; y) = x4 + 2y2 � 2xy

6. Find extreme points for the function f(x; y) = x2 + y2 subject to x2 + xy + y2 = 3:

7. Find the optimal paths of the control, state, and co-state variables to

min2R0

(t u+ u2)dt

subject to�x = x+ ux(0) = 1; x(2)� freeu(t)� unconstrained

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8. Find the optimal paths of the control, state, and co-state variables to

max1R0

(x+ u) dt

subject to�x = �x+ ux (0) = 4; x (1) -freeu (t) 2 h0; 1i

Be sure to check that the Hamiltonian is maximized rather than minimized.

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