Find f(6) if f(x) = x^{2} ÷ 3 + x.
1 answer:
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The first equation is linear:
Divide through by
to get
and notice that the left hand side can be consolidated as a derivative of a product. After doing so, you can integrate both sides and solve for
.
![\dfrac{\mathrm d}{\mathrm dx}\left[\dfrac1xy\right]=\sin x](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20d%7D%7B%5Cmathrm%20dx%7D%5Cleft%5B%5Cdfrac1xy%5Cright%5D%3D%5Csin%20x)
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The second equation is also linear:
Multiply both sides by
to get
and recall that
, so we can write
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Yet another linear ODE:
Divide through by
, giving
![\dfrac{\mathrm d}{\mathrm dx}[\sec x\,y]=\sec^2x](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20d%7D%7B%5Cmathrm%20dx%7D%5B%5Csec%20x%5C%2Cy%5D%3D%5Csec%5E2x)
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In case the steps where we multiply or divide through by a certain factor weren't clear enough, those steps follow from the procedure for finding an integrating factor. We start with the linear equation
then rewrite it as
The integrating factor is a function
such that
which requires that
This is a separable ODE, so solving for
we have
and so on.
Divide through by
and notice that the left hand side can be consolidated as a derivative of a product. After doing so, you can integrate both sides and solve for
- - -
The second equation is also linear:
Multiply both sides by
and recall that
- - -
Yet another linear ODE:
Divide through by
- - -
In case the steps where we multiply or divide through by a certain factor weren't clear enough, those steps follow from the procedure for finding an integrating factor. We start with the linear equation
then rewrite it as
The integrating factor is a function
which requires that
This is a separable ODE, so solving for
and so on.