2
$\begingroup$

I have an equation like this:

$te^{t} = \int\nolimits_0^t e^\tau u(\tau)d\tau$

I don't really know how to solve it.. Would it be possible to differentiate both sides of the equation? If so, how can I do it - is it like differentiate one side and then the other or is it more complex? And if it is possible what conditions need to be satisfied in order to do it?

  • 0
    You were very close to having the right syntax. All I did was change your ampersands (&) to double dollar signs ($$).2011-02-17

1 Answers 1

4

If we differentiate the left hand side we get $e^t+t e^t$, if we differentiate the right hand side we get $e^t u(t)$ by the fundamental theorem of calculus. Then we see

$u(t)=t+1$

An we are done. I think we need to demand that $u(t)$ is continuous to get uniqueness. In general $u(t)=t+1$ almost everywhere.

  • 1
    Before reading this comment I tried differentiating using both F(b) and F(a). It did produce a solution: $1 + t + \frac{u(0)}{e^t}$ so I checked if it was ok and found out that u(0) needs to be equal to 0 in order to satisfy the initial equation. Thank you all for help.2011-02-17