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I have been given the following first order derivative equation:

$$\frac{dx}{dt}=ax-bxy$$

How do I find the second derivative? I know how to find implicit derivatives but since the differential in this case needs to be found in terms of $dx/dt$, instead of the $dy/dx$ which I am used to, I am confused. Should I used the product rule to find the derivative of $-bxy$?

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  • $\begingroup$ y is a function of t? $\endgroup$
    – Srini
    Jan 17 '18 at 1:08
  • $\begingroup$ x and y are both functions of t, start there and remember you know $\frac{dx}{dt}$ $\endgroup$ Jan 17 '18 at 1:50
  • $\begingroup$ @Srini, yes y is a function of t. $\endgroup$ Jan 17 '18 at 2:18
  • $\begingroup$ @JacobClaassen Okay, I will try figuring it out! $\endgroup$ Jan 17 '18 at 2:19
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$$\frac{dx}{dt}=ax-bxy$$

You know implicit differentiation, so let's just differentiate our above equation wrt $t$ (assuming $a,b$ constants):

$$\frac{d^2x}{dt^2}=a\frac{dx}{dt}-b\frac{d(xy)}{dt}$$

Now use the product rule to find the differentiation of $xy$ wrt $t$. You'll get $dx/dt$ again, but remember to substitute its values from the original expression we have. You'll also get $dy/dt$, but you cannot further simplify as we don't know exactly how $y$ is a function of $t$.

That's all the logic there is to your question. Hope it helps!

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  • $\begingroup$ Lovely, thanks! :) $\endgroup$ Jan 17 '18 at 2:32
  • $\begingroup$ @SelenaCarlos You're welcome :) $\endgroup$ Jan 17 '18 at 2:32

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