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You don't have the plus or minus, right.
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And we sort of know in every problem
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there must be two solutions,
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because we should be able to pick the initial position
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and velocity at will.
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If a solution has only one free parameter,
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you cannot pick two numbers at will.
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Now, that's a piece of mathematics I don't want to do now,
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but what you can show is in that case,
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the second solution looks like this.
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It's a new function, not an exponential
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but t times an exponential will solve the equation.