Math 305  Homework Set 1  Spring 2017

Due Monday, April 10

I. Find the general solution to each of the following.

1. (e^{x} sin y − 2y sin x) + (e^{x} cos y + 2 cos x)y′ = 0.

2. t^{2}y′ + 2ty − y^{3} = 0.

3. y′ − 2y = 4 − x.

4. 2y′ − y = e^{x/3}.

II. Find all solutions to the following boundary value problems (this is based on Section 10.1).

1. y′′ + y = 0 with y(0) = 0, y(π) = 0.

2. y′′ + y = 0 with y(0) = 0, y(π/3) = 0.

3. y′′ + 3y = 0 with y(0) = 0, y(π) = 0.

4. For which values of γ will y′′ + γy = 0, y(0) = 0, y(π) = 0 have nontrivial solutions? Explain.

5. y′′ + y = 0 with y′(0) = 0, y′(π) = 0.

More on next page.

III. Find the Fourier series of each of the functions shown below. (Make sure you are using the correct value for L.) You can do the integration with a computer or calcuator, but indicate when you have done so. Write out at least the first 10 terms.

1.

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2 4 6 8 10 −2 −4 −6 −8 −10

2.

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2 4 6 8 10 −2 −4 −6 −8 −10

3.

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1 2 3 4 5 −1 −2 −3 −4 −5

III. For #1 above use a compute to plot the partial sums for N = 1, 2, 5, and 10.