18.01 PROBLEM SET 4

Hand in to Room 2-108 by 1:00 pm Friday, March 22.

The due date and time are absolute. Late homeworks are accepted until the solutions are posted, but the score will be multiplied by 1/2.

Everything included here may be used later. You do not need to memorize the proofs, but you need to know the facts.

1. MVT.1/3, 5. 135/10, 18, 19. 141/21, 24. 144/6, 11, 19.

(10 points)

2. In this exercise we solve the differential equation f′′(x) = f(x). In other words we are looking for functions which are their own second derivatives. As a warm up, first find all of those g functions which satisfy g′(x) = g(x).

(a) Suppose g′(x) = g(x). Prove that there exists a c_{2} constant such that

g(x) = c_{1} · e^{x}.

(b) Suppose h′(x) = −h(x). Prove that there exists a c_{2} constant such that

h(x) = c_{2} · e^{−x}.

(c) Suppose f′′(x) = f(x). Prove that there exist c_{1}, c_{2} constants such that

f(x) + f′(x) = c_{1} · e^{x} and f(x) − f′(x) = c_{2} · e^{−x}.

(d) Conclude that there exist a, b constants such that

f(x) = a · e^{x} + b · e^{−x}.

Remark. Note that sinh x and cosh x are their own second derivatives and indeed they are among the solutions, too.

(10 points)

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