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Math Help - verify that the pair of functions is a solution...

  1. #1
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    verify that the pair of functions is a solution...

    verify that the pair of functions x = t + e^t , y = te^t is a solution to the system of differential equations.

    x + y - dy/dt = 1 , x - dx/dt = t - 1.

    ok so i was able to prove that the pair of functions were solutions to the x + y - dy/dt = 1.

    but i couldnt prove they were solutions to the equation on the right, x - dx/dt = t - 1.

    any thoughts? thanks in advance.
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  2. #2
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    to do this problem i substituted the two equations into the x + y - dy/dt = 1. and solved it for dy/dt, the problem checked out. but i couldnt do that with the other side?? any thoughts?
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  3. #3
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    \displaystyle x = t + e^t, so \displaystyle \frac{dx}{dt} = 1 + e^t.

    What does \displaystyle x - \frac{dx}{dt} equal?
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  4. #4
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    sorry, X - dx/dt = t - 1
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    Exactly...

    \displaystyle x - \frac{dx}{dt} = t + e^t - (1 + e^t) = t - 1.


    Therefore we have verified \displaystyle x = t + e^t is a solution to that DE.
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  6. #6
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    ok but what about the function with the y in it? wouldnt i have to prove that y=te^t is also a solution of x-dx/dt = t-1 ?
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  7. #7
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    How can it be? There's no \displaystyle y in it...
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    thats what i was hoping. my prof wanted me to prove that both were solutions. thank you!
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  9. #9
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    Quote Originally Posted by slapmaxwell1 View Post
    thats what i was hoping. my prof wanted me to prove that both were solutions. thank you!
    What I mean is, in the second DE \displaystyle y can be anything, as the DE doesn't depend on it.

    So to have a solution to both DEs, you just need a \displaystyle y that satisfies the first...
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