# Help with a differential word problem

When the electromotive force (emf) is removed from a circuit containing inductance and resistance but no capacitors, the rate of decrease of current is proportional to the current. If the initial current is 30 amps but decays to 11 amps after 0.01 seconds, find an expression for the current as a function of time.

Do I use the $y^{\prime}=-ky$ ? or $y^{\prime}=\dfrac{-k}{y}$

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If $a$ is proportional to $b$ then $a = kb$. (With constant $k$ possibly negative). –  in_wolframAlpha_we_trust Feb 22 '13 at 7:25
@in_wolfram_we_trust what do you think of my answer –  yiyi Feb 22 '13 at 8:38

THanks to @in_wolfram_we_trust.

This is my solution, hope its correct.

Using $\dfrac{dy}{dt} = -ky$ with the given conditions $y=30, t = 0$ and $y = 11, t = 0.01$

$\dfrac{\mathrm{d}y}{\mathrm{d}t} = -ky$ collect the varibles together.

$\int \frac{\mathrm{d}y}{y} = \int -k\; \mathrm{d}t$

results to $y = Ce^{-kt}$

$30 = Ce^{-k\left(0\right)} \Rightarrow C = 30$

$11 = 30e^{-k\left(0.01\right)} \Rightarrow -k = \dfrac{\ln{\vert 11/30 \vert}}{0.01} \Rightarrow k = 100.3$

Thus the equation is $y=30e^{-100.33t}$

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That looks good. –  in_wolframAlpha_we_trust Feb 22 '13 at 8:56