Thread: Problem 24
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Old May 30th, 2007, 08:29 PM
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Okay, I oopsied. But still:

f(x) = Ae^x

Thus
f(t) = Ae^t

\frac{df}{dt} = Ae^t

Thus
\int_0^x dt \, [ f(t) ]^2 + \left [ \frac{df}{dt} \right ] ^2

= \int_0^x dt \, (A^2e^{2t} + A^2e^{2t} )

= \int_0^x dt \, 2A^2e^{2t}

= 2A^2 \int_0^x dt \, e^{2t}

= 2A^2 \frac{1}{2}e^{2t}|_0^x

= A^2(e^{2x} - 1)

And
[ f(x) ] ^2 = A^2e^{2x}

So
[ f(x) ] ^2 \neq \int_0^x dt \, [ f(t) ]^2 + \left [ \frac{df}{dt} \right ] ^2

How can this then be the solution of the integral equation? I'm missing something here...

-Dan
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