MATH 3410 Concept of Probability Problems

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MATH3410 Final Practice I
Problem I True or False, no need to explain your answer.
1. Any solution of x0 = arctan x + t cannot have maximal points.
2. Any solution of x0 = sin(tx) must be an even function.
3. If the Wronskian of two functions f (x), g(x) is zero for all x, then f and g are linearly
dependent.
4. RFor any positive integers m 6= n, then sin(mx) and cos(nx) are orthogonal in that
π
sin(mx) sin(nx)dx = 0.
0
5. If x(t) is a solution to x00 + 3×0 + 2x = 0, then limt→+∞ x(t) = 0.
Problem II Solve the heat equation
ut − uxx = 0 x ∈ [0, π], t ∈ [0, ∞)
with initial condition u(0, x) = sin x, and boundary condition u(t, 0) = 0, ux (t, π) = 0.
Problem III Consider the following 2 × 2 system
(
x0 = 2x + 3y
y 0 = −3x + αy
1. Find the value α so that the system above is Hamiltonian
2. If (x(t), y(t)) is a solution with starting point (0, 1), is it periodic? State your reason.
Problem IV Consider the second order nonlinear equation
x00 − x − 8×7 = 0 x(0) = −1, x0 (0) =

3.
Transfer the equation to a 2 × 2 hamiltonian system and then find limt→+∞ x(t).
Problem V Find the power series which solves the Cauchy problem
(
x00 (t) = tx + 1
x(0) = 1, x0 (0) = −1
Problem VI
Use Frobenius method to solve
tx00 − (t + 3)x0 + x = 0.
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Tags:
mathematics

differential equation

random variables

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