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Math Help - sketch an approximate solution curve through each of the points...

  1. #1
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    sketch an approximate solution curve through each of the points...

    by hand sketch an approximate solution curve through each of the points.

    ok so the function is:

    y(dy/dx) = -x so, (dy/dx) = -x/y at the point y(1)= 1 and y(0) = 4

    so at the point 1,1 the slope is -1, so my dash is going up and to the left? is this all im doing with this problem or am i missing something?

    thanks in advance.
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  2. #2
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    It's well known that circles have a derivative of \displaystyle -\frac{x}{y}.

    You can check this by solving the DE...

    \displaystyle y\,\frac{dy}{dx} = -x

    \displaystyle \int{y\,\frac{dy}{dx}\,dx} = \int{-x\,dx}

    \displaystyle \int{y\,dy} = -\frac{x^2}{2} + C_1

    \displaystyle \frac{y^2}{2} +C_2 = \frac{x^2}{2} + C_1

    \displaystyle \frac{x^2}{2} + \frac{y^2}{2} = C_1 - C_2

    \displaystyle x^2 + y^2 = r^2 where \displaystyle r^2 = 2(C_1 - C_2).


    Now you can use your boundary conditions to evaluate \displaystyle r^2.
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