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Find a function that satisfies the following Differential Eq

  1. Jun 17, 2017 #1
    I'm helping some guys with Calculus I class and found this exercise in the practice about integrals.
    I think it's overkill but it may have some easy way to solve it.
    I'm very rusty solving differential equations.

    1. The problem statement, all variables and given/known data

    Find f differentiable such that
    $$
    (3+f'(x))e^{2-x} = (x-6) (3x+f(x))^{2}
    $$
    with f(2)=0.

    2. Relevant equations
    I have solved similar exercises searching a function g(x) such that g(f(x))'=g'(f(x)).f'(x) and and putting the equation in a way that you have g'(f(x))=h(x). So you can integrate and solve it. The most classic example is (ln(f(x))'=f'(x)/f(x), but I have done with more complicated functions. In this case I had to go to wolframalpha because I couldn't figure out the solution, and even seeing the solution I'm not having an easy time finding it, I only was able to verify that's a solution. If it were a linear differential equation it would be easier, but the square spoils everything.

    3. The attempt at a solution
    I also thought about integrating by parts the left part of the equality, but I'm not sure if it will help to arrive an easier equation as I don't know what to do with the expressión on the right.

    Any suggestions? I'm very rusty with differential equations but as this exercise was in a calculus 1 class I think it should be easier to solve.

    Thank you for your help!
     
  2. jcsd
  3. Jun 17, 2017 #2

    ehild

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    Try to solve it for the function g(x) = 3x+f(x)
     
  4. Jun 17, 2017 #3
    Let's see if I got it.
    Let g(x)=f(x)+3x. Then g'(x)=f'(x)+3.
    So the equation becomes
    $$
    g'(x).e^{2-x} = (x-6) g(x)^{2}
    $$
    We work the expression
    $$
    \dfrac{g'(x)}{ g(x)^{2} } = (x-6) e^{x-2}
    $$
    We re-write the expression on the left
    $$
    -( \dfrac{1}{g(x)} )' = \dfrac{g'(x)}{ g(x)^{2} } = (x-6) e^{x-2}
    $$
    So
    $$
    ( \dfrac{1}{g(x)} )' = (6-x) e^{x-2}
    $$
    Now all we have to do is to integrate and find the g(x) (and so f(x))
     
  5. Jun 17, 2017 #4

    ehild

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    It will be all right.
     
  6. Jun 17, 2017 #5
    Thank you for your help. Also, your signature is great, having at hand those symbols is very convenient.
     
  7. Jun 17, 2017 #6
    All ready answered while I was typing.
     
    Last edited: Jun 17, 2017
  8. Jun 17, 2017 #7

    ehild

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    You find those symbols and a couple of more by hitting the ∑ button above if you write a post. .
     
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