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Math Help - Quick problem I need a hand with please!

  1. #1
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    Question Quick problem I need a hand with please!

    I have this question as an assignment, puzzled with one bit...

    let y = Q(x) be any solution to the differential equation.

    dy/dx = xy

    Answer each of the following without solving this DE.

    (i) In terms of x only, what is the slope of Q as it crosses the line y = x ?

    (ii) What is the slope of the line y = x ?

    (iii) " In the region x > 1, the graph of Q can only cross the line y = x from below" Is this statement true or false??

    My attempt...

    (i)= slope = dy/dx = xy = x . x = x2

    (ii)= 1

    (iii)= TRUE = but the only way i found it to be true is by solving the DE and graphing it which is not allowed by the question. Am i right firstly in saying it is true and if so how would I know this without solving DE??

    Thanks alot!

    Nappy
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  2. #2
    Member Mauritzvdworm's Avatar
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    The answers to the first two questions are correct, now just use those to answer the third question

    You know that when x=y we have \frac{dy}{dx}=x^{2}, so when we consider what is happening of the line we know that y=\frac{1}{3}x^{3}+c

    when you draw the graph of these equations you will see that it is true
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  3. #3
    MHF Contributor

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    Quote Originally Posted by Nappy View Post
    I have this question as an assignment, puzzled with one bit...

    let y = Q(x) be any solution to the differential equation.

    dy/dx = xy

    Answer each of the following without solving this DE.

    (i) In terms of x only, what is the slope of Q as it crosses the line y = x ?

    (ii) What is the slope of the line y = x ?

    (iii) " In the region x > 1, the graph of Q can only cross the line y = x from below" Is this statement true or false??

    My attempt...

    (i)= slope = dy/dx = xy = x . x = x2

    (ii)= 1

    (iii)= TRUE = but the only way i found it to be true is by solving the DE and graphing it which is not allowed by the question. Am i right firstly in saying it is true and if so how would I know this without solving DE??

    Thanks alot!

    Nappy
    Simpler, I think. You have already said that, on this line, dy/dx= x^2 which is never negative. That is, the graph must cross the line from below because it cannot be decreasing there.
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  4. #4
    Junior Member
    Joined
    Jan 2010
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    Question

    Thanks guys,

    think ive gotten myself a little confused though. Although the slope can never be negative and is always increasing could it not be above the line y = x. I solved for the curve and got

    y = 1/3 x3 + c

    Since this has infinitely many solutions, wouldnt a high constant of integration push it above the line making the statement false or am I losing the plot here??

    Thanks again....
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