Mathematics problems

Final Project MATH 311

1. Consider the ODE x′′ − 3x′ + 2x = 0 (1)

This problem will solve this ODE in two different ways.

a. Define v = x′. By writing x′′ = v dv dx , convert this ODE into a first order ODE for the

velocity v as a function of position x.

b. Notice that this equation is homogenous. Convert the ODE you derived in part (a) into a separable equation using the change of unknowns u(x) = v(x)/x.

c. Integrate the equation you found in part (b) assuming the initial condition v(x = 2) = 3. Your answer should be an equation relating v and x.

d.. Solve this equation using methods from Chapter 2 and explain how the relation you derived in part (c) confirms this solution.

2. A chain partly rests on the top of a cube measuring 20 meters to a side so that x meters of the chain hang over the edge. When the force due to gravity acting on the overhanging portion of the chain as well as friction is taken into account, the velocity v of the chain hanging over the side of the cube is modeled by the ODE

20v dv

dx = gx − (20 − x)v2 (2)

where v = v(x) is the velocity of the chain as a function of the overhang x, and g is the constant acceleration due to gravity.

a. Use the substitution w = v2 to convert this ODE into a linear ODE.

b. Solve this linear ODE assuming that the chain is at rest when 2 meters of chain are hanging over the side of the cube.

3. Consider the ODE x′′ + ω2x = 0. (3)

a. Find all possible solutions to this ODE when ω = 1 that satisfy the boundary conditions x(0) = x(2π) = 0.

b. Find all possible solutions to this ODE when ω = n is a positive integer satisfying the boundary conditions x(0) = x(2π) = 0.

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c.. Explain why the only solutions to this problem when ω2 is not an integer is x(t) = 0.

4. Use the Laplace Transform to derive the solution to the initial value problem

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(1 − t)y′′ + ty′ − y = 0 y(0) = 0

y′(0) = 1

(4)

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