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Unformatted text preview: 18.03 Problem Set 1 Fall 2009 Solutions 1. (5 points) Notes 1A5(d). For what values of t is the solution x ( t ) defined? What happens to x ( t ) as t tends to ±∞ ? The original ODE is dx dt = √ 1 + x t 2 + 4 . ODE There is a solution x ( t ) = − 1 defined for all t that is not mentioned in the solutions in the notes. So for the rest of the problem we can assume that x ( t ) is (at least sometimes) not equal to − 1. Just as explained in the solution in the notes, you can then separate the variables x and t , writing dx √ 1 + x = dt t 2 + 4 . (Since x ( t ) negationslash = − 1, we’re (at least sometimes) not dividing by zero.) Taking antiderivatives of each side is easy, and we get 2 √ 1 + x = 1 2 tan 1 ( t/ 2) + c. Integrated Here we’re using the “principal value” of the inverse tangent, that takes values strictly between − π/ 2 and π/ 2. Solving for x gives x ( t ) = 1 4 parenleftbigg 1 2 tan 1 ( t/ 2) + c parenrightbigg 2 − 1 . GeneralSolution ? Now we need to ask whether this is actually a solution of the original differential equation. Everything that I did was legal and reversible as long as x ( t ) > 1, except the squaring that I did to solve (Integrated) for x ( t ). Since the square root sign means the nonnegative square root, equation (Integrated) can be true only when the right side is nonnegative. By trigonometry, 1 2 tan 1 ( t/ 2) + c ≥ if and only if braceleftbigg c ≥ π/ 4 , or − π/ 4 < c < π/ 4 , and t ≥ 2 tan( − 2 c ) . If you think about this a little more, you’ll see that the complete list of solutions of (ODE) is the boring solution x ( t ) = − 1; the solutions x ( t ) = 1 4 parenleftbigg 1 2 tan 1 ( t/ 2) + c parenrightbigg 2 − 1 ( c ≥ π/ 4) , SomeSolutions and the strange solutions x ( t ) = braceleftBigg 1 4 ( 1 2 tan 1 ( t/ 2) + c ) 2 − 1 ( t ≥ 2 tan( − 2 c )) − 1 ( t ≤ 2 tan( − 2 c )) . StrangeSolutions defined for − π/ 4 < c < π/ 4. You should try to understand why the boring solution and the strange solutions (which overlap for long periods of time) do not contradict the uniqueness theorem on page 24 of the text....
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This note was uploaded on 05/06/2010 for the course 18 18.03 taught by Professor Unknown during the Spring '09 term at MIT.
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