How to reduce?

  1. 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 !
     
  2. jcsd
  3. arildno

    arildno 12,015
    Science Advisor
    Homework Helper
    Gold Member

    Evaluate your equation at t=0. What does that say about u(0)?

    Differentiate your equation to get your differential equation.
     
  4. 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?
     
  5. lurflurf

    lurflurf 2,326
    Homework Helper

    Use the Leibniz Integral Rule.
    [tex]\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)}[/tex]
    [tex]\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)}[/tex]
    http://mathworld.wolfram.com/LeibnizIntegralRule.html
    for the initial condition you should know that
    [tex]\int_0^0 f(x) dx=0[/tex]
     
    Last edited: Aug 23, 2005
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