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Show that $$F(f)(t) = t^2 + \frac{t}{3}f(t) + \frac{1}{5}\int_0^t e^uf(u) du$$ is a contraction on $(C[0, 1), d_u)$.

Deduce that the differential equation $$(15 − 5t)\frac{df}{dt} = (5 + 3e^{t})f + 30t$$ has a unique solution in $C[0, 1]$.

Answer I have completed the first part and shown that $F$ is a contraction on $C[0, 1]$. Then by the contraction mapping theorem there exists a fixed point $f \in C[0, 1]$ such that $F(f)(t) = f(t)$

So I have $f(t) = t^2 + \frac{t}{3}f(t) + \frac{1}{5}\int_0^t e^uf(u) du$

and if I rearrange and differentiate I get $$(15 − 5t)\frac{df}{dt} = (5 + 3e^{t})f + 30t$$

So I know $f$ is a solution to the differential equation. But I have to now show it is the only solution. Normally that is done by taking another function, $g$, from $C[0, 1]$ that satisfies the equation and showing that it must equal $f$. However I am unable to make that work here.

And in fact, I don't understand how there can be a unique solution anyway when we are given no initial condition such as $x(0) = 1$...and then if we had $f(t) = t^2 + \frac{t}{3}f(t) + \frac{1}{5}\int_0^t e^uf(u)du$ + K, where $K$ is any constant, this will also satisfy the equation. So is this a trick question and the differential equation doesn't ac

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    The contraction has a _unique_ fixed point.2012-11-19
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    Yes and this is used to show that a solution does exist for the differential equation. But I need to show that this solution is unique.2012-11-19
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    The fixed point _is_ the solution ;)2012-11-19
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    Yes, it is the solution. But it has to be shown that it is the only solution. For instance, if I had $f(t) = t^2 + \frac{t}{3}f(t) + \frac{1}{5}\int_0^t e^uf(u) du + 9999999999$ I could rearrange and differentiate (which would get rid of the constant), and it would satisfy the differential equation. So why is it that $f(t) = t^2 + \frac{t}{3}f(t) + \frac{1}{5}\int_0^t e^uf(u) du$, i.e. with no constant, is the unique solution to the equation?2012-11-19
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    No is not, as it doesn't satisfy the initial condition. The solution is _unique_ when initial conditions are imposed. You should really study the [Piccard's Theorem](http://en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem). The uniqueness _is consequence_ of the contraction. As the operator is a contraction, the iterations converge to the fixed point, which is unique, i.e., they cannot converge to any other point. Also, the addition of the constant works because _this ode_ is linear; Piccard's is more general and it works whit nonlinear odes, where you cannot add constants blindly.2012-11-19
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    [Here](http://student.fizika.org/~aficnar/Korisno/Picard.pdf) are some decent notes on the subject. I recommend you to read Braun's _Differential Equations and Their Applications_ for thourough (introductory level) proofs of existence and uniqueness theorems.2012-11-19
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    What is the initial condition in this case? For 'regular' differential equations I see initial conditions like $y(0) = 1$, so in this case I'd be expecting something like $F(f(0)) = ...something$, but there is nothing specified. So what is the initial condition?2012-11-29
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    how did you prove contraction?2013-11-22

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