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Name: Perm number:

Midterm – in-class part

Time: 75 minutes 1. (25 points)

• Find all real solutions to y00+ y = x2.

• Find all real solutions to y00+ y = x2 for which y(0) = 0 and y0(0) = 0.

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2. (25 points) Consider the equation

y000− y00+ y0− y = 0

• What are all of its real solutions?

• What are all of its real solutions for which y(0) = 0, y0(0) = 1, and y00(0) = 0?

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3. (25 points) Show how to derive a power series solution to y00+ 2y = 0 for which y(0) = 1 and y0(0) = 0. List terms up to x4.

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4. (25 points) Suppose the second order linear ODE

y00+ p(x)y0+ q(x)y = 0 has two solutions f (x) and g(x) and you know that the Wronskian

W (f (x), g(x)) > 0 for all x. Using f (x) and g(x) explain how to find a solution h(x) of

y00+ p(x)y0+ q(x)y = 0 for which h(1) = 3 and h0(1) = 7.

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