# how to reduce?

by yukcream
Tags: reduce
 P: 60 How to reduce the equation: Show how the equation u(t)+Int( 0->t) {[e^a(t-t1)]u(t1)}dt1 = k can be reduced to a differential equation and obtain an intial condidtion for the equation. Remarks: Int(0->t): integral from 0 to t !
 PF Patron HW Helper Sci Advisor Thanks P: 11,935 Evaluate your equation at t=0. What does that say about u(0)? Differentiate your equation to get your differential equation.
P: 60
 Quote by arildno Evaluate your equation at t=0. What does that say about u(0)? Differentiate your equation to get your differential equation.
Yes I know I have to differential but if the diff. it with respect to t , what will happen to the second term on the left hand side?

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P: 2,078

## how to reduce?

 Quote by yukcream Yes I know I have to differential but if the diff. it with respect to t , what will happen to the second term on the left hand side?
Use the Leibniz Integral Rule.
$$\frac{d}{dt}\int_{a(t)}^{b(t)}f(x,t)dx=\int_{a(t)}^{b(t)} \partial_tf(x,t)dx+f(x,t)\partial_tx|_{x=a(t)}^{x=b(t)}$$
$$\frac{d}{dt}\int_{a(t)}^{b(t)}f(x,t)dx=\int_{a(t)}^{b(t)} \frac{\partial}{\partial t}f(x,t)dx+f(x,t){\partial x}{\partial t}\right|_{x=a(t)}^{x=b(t)}$$
http://mathworld.wolfram.com/LeibnizIntegralRule.html
for the initial condition you should know that
$$\int_0^0 f(x) dx=0$$

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