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Homework Help: Diff EQ Intro - Verify Family of Functions as Solution

  1. Aug 27, 2010 #1
    1. The problem statement, all variables and given/known data
    Verify that the indicated family of functions is a solution of the given differential equation. Assume an appropriate interval I of definition for each solution.


    2. Relevant equations
    dy/dx + 2xy = 1; y = e-x2[tex]\int[/tex](from 0 to x)et2dt + c1e-x2


    3. The attempt at a solution
    I have only had one class period in differential so far and we didn't get to go over much material. I imagine that one would need to differentiate y(x) with respect to x and plug into the first equation. However, I'm not quite sure what to do with the integral with respect to t. I tried to integrate it, and got et2/(2t), but evaluating that at 0 would cause an implosion. If I differentiate with respect to x, I don't think I can just treat it as a constant because it's evaluated from 0 to x. Could I please get a nudge in the right direction?
     
  2. jcsd
  3. Aug 27, 2010 #2
    This is just simple application of Fundamental Theorem of Calculus.

    If [tex] F(x) = \int_a^x f(t) dt [/tex] then [tex] F'(x) = f(x), [/tex] given of course that f(x) is continuous on [a, x].
     
  4. Aug 28, 2010 #3
    So then dy/dx would be:

    dy/dx = e-x2 * ex2 - 2xe-x2[tex]\int[/tex](from 0 to x)et2dt - 2xc1e-x2

    And then plugging it into the differential equation, it all cancels out. Thank you so much!
     
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