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Find and solve a differential equation for the decay of a material with given properties

by RoRi

A given substance decays at a rate proportional to the squareroot of the amount of the material present. Show that a substance with this decay property will completely decay in a finite amount of time and find this time.

Incomplete.

One comment

  1. Evangelos says:

    As the problem mentions, instead of the rate of decay being proportional to the square of the amount of material present, the rate of decay is proportional to the square root of the amount present. Our equation looks like this:

        \begin{align*} \frac{dy}{dt} &= -k\sqrt{y} \end{align*}

    Where k is a constant.

    As before, this is a separable equation. Separating variables and integrating from time 0 to some arbitrary time x>0 (unfortunately, no backwards time travel here) gives us:

        \begin{align*} -\int_{y(0)}^{y(x)} \frac{dy}{\sqrt{y}} &= k\int_{0}^{x} dt \end{align*}

        \begin{align*} 2[\sqrt{y(0)} - \sqrt{y(x)}] &= kx \end{align*}

    And since we know the values for both y(0) and y(1) from our given information, we can find an explicit value for k

        \begin{align*} y(0) &= 1 \\ y(1) &= \frac{1}{2} \end{align*}

        \begin{align*} 2 - \sqrt{2} &= k \end{align*}

    Back to our equation for y(x), with our newly found value for k, along with our given information, we have

        \begin{align*} 2(1 - \sqrt{y(x)}) &= (2 - \sqrt{2})x \end{align*}

        \begin{align*} [1 - \frac{1}{2}(2 - \sqrt{2})x]^{2} &= y(x) \end{align*}

    And, we wish to show that there is some value of x such that y(x) = 0. In other words:

        \begin{align*} [1 - \frac{1}{2}(2 - \sqrt{2})x]^{2} &= 0 \end{align*}

    This is satisfied by the following

        \begin{align*} x &= \frac{2}{2 - \sqrt{2}} \\ \\ &= 2 + \sqrt{2} \end{align*}

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