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Math Help - Combining two differential equations

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    Combining two differential equations

    Two quantities, x and y, vary with time t. State the differential equation for each of the following.
    (a) The rate of change of y with respect to t varies inversely as x.
    (b) The rate of change of x with respect to t varies inversely as y. Combining the differential equation of (a) and (b), form a differential equation involving only x and y, solve the differential equation expressing y in terms of x.

    How do I combine the two differential equations into one?
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    Re: Combining two differential equations

    Quote Originally Posted by alexander9408 View Post
    Two quantities, x and y, vary with time t. State the differential equation for each of the following.
    (a) The rate of change of y with respect to t varies inversely as x.
    (b) The rate of change of x with respect to t varies inversely as y. Combining the differential equation of (a) and (b), form a differential equation involving only x and y, solve the differential equation expressing y in terms of x.

    How do I combine the two differential equations into one?
    Do you understand what's meant by inverse proportion? If "a" is inversely proportional to "b", then \displaystyle a = \frac{k}{b}, where "k" is a constant.
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    Re: Combining two differential equations

    Yes, my problem was combining the two differential equations dy/dt = k/x and dx/dt = h/y (k and h are constants) into one differential equation and expressing y in terms of x.
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    Re: Combining two differential equations

    Remember \displaystyle \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{dy}{dx}, so

    \displaystyle \begin{align*} \frac{\frac{dy}{dt}}{\frac{dx}{dt}} &= \frac{\frac{k}{x}}{\frac{h}{y}} \\ \frac{dy}{dx} &= \frac{k\,y}{h\,x} \end{align*}

    This is a separable first order DE.
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