17 3/4 - 2/5x = 4 1/2
-2/5x = 4 1/2 - 17 3/4
-2/5x = 9/2 - 71/4
-2/5x = 18/4 - 71/4
-2/5x = - 53/4
x = (-53/4) / (-2/5)
x = -53/4 * - 5/2
x = 265/8
x = 33 1/8 bottles
Suppose
is another solution. Then

Substituting these derivatives into the ODE gives


Let
, so that

Then the ODE becomes

and we can condense the left hand side as a derivative of a product,
![\dfrac{\mathrm d}{\mathrm dx}[x^5u]=0](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Cmathrm%20d%7D%7B%5Cmathrm%20dx%7D%5Bx%5E5u%5D%3D0)
Integrate both sides with respect to
:
![\displaystyle\int\frac{\mathrm d}{\mathrm dx}[x^5u]\,\mathrm dx=C](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cint%5Cfrac%7B%5Cmathrm%20d%7D%7B%5Cmathrm%20dx%7D%5Bx%5E5u%5D%5C%2C%5Cmathrm%20dx%3DC)

Solve for
:

Solve for
:

So another linearly independent solution is
.
Go to a website called desmos it’s a graphing calculator and you just type what you want graphed and it shows it
The answer for you problem is 6