DiffEqReview - Practice Exercises on Differential Equations...

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Practice Exercises on Differential Equations What follows are some exerices to help with your studying for the part of the final exam on differential equations. In this regard, keep in mind that the exercises below are not necessarily examples of those that you will see on the final exam. Even so, if you understand how to do these, you should do fine on the differential equation portion of the final. The answers are provided at the end. Exercises: 1. Find the Fourier series of the function on [- π , π ] that equals x where x 0 and zero where x < 0. 2. Find the Fourier series of the function on [- π , π ] given by x |sin(x)|. 3. Let f(x) denote a function on [- π , π ] with the property that f(x) = f(-x) for all x. Explain why there are no sine functions in the Fourier series of f. 4. By its very definition, the Fourier series of a smooth function x f(x) on [- π , π ] has the form f(x) = a 0 + k=1,2,… (a k cos(kx) + b k sin(kx)). When computing the Fourier series of the derivative, f´(x), there is the inevitable temptation to exchange orders of differentiation and summation and so conclude that f´(x) has the Fourier series k=1,2,… (k b k cos(kx) - k a k sin(kx)). Show that this is the correct answer when f( π ) = f(- π ) by computing the relevant integrals. 5. Find a basis for the kernel of the linear operator f f´´ + 3f´ – 4f on the space of smooth functions on [0, 1]. Find an element in the kernel of this operator that obeys f(0) = 1 and f(1) = -1. 6. Find a basis for the kernel of the linear operator f f´´ + 4f on the space of smooth functions on [- π , π ]. Find an element in the kernel of this operator that obeys f(0) = 1 and f´(0) = -1. 7. An inner product on the space of continuous functions on [- π , π ] is defined as in the Differential Equation Handout using the rule that has the inner product between functions f and g equal to 1 ! !" " # f(x)g(x) dx. Exhibit a non-zero function, g, and an infinite, orthonormal set such that g is orthogonal to each element in this set. 8. Let A denote the linear operator f f´ and let B denote the linear operator f x f,
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where f here is any smooth function on (- , ). What is the operator AB – BA? 9. A ubiquitous operator in quantum mechanics sends a function, f, on (- , ) to the function T(f) = –f´´ + x 2 f – f. a) Suppose that f is a function that obeys f´+ x f = 0. Prove that T(f) = 0. b) Write down all functions f that obey f´ + x f = 0. 10. Reintroduce the inner product from Problem 7 on the space of functions in [- π , π ]. Let f be any function on [- π , π ] that vanishes at the endpoints. Write the orthogonal projection of f onto the span of {1, x} as a + bx. Give the orthogonal projection of f´(x) onto the span of {1, x, x 2 } in terms of a and b. 11. Let x f(x) be a smooth function such that f(1) = 2 while f(x) < 2 if x 1. Let c > 0 be a constant.
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DiffEqReview - Practice Exercises on Differential Equations...

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