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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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Explanation & Answer:

5 Problems

Tags:

mathematics

differential equation

random variables

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