Alan

3.8*10⁸/1.2*10¹⁵ = (3.8/1.2)*10⁸*10^-15 = (3.8/1.2)*10^8-15 = (3.8/1.2)*10^-7

$$\left({\frac{{\mathtt{3.8}}}{{\mathtt{1.2}}}}\right){\mathtt{\,\times\,}}{{\mathtt{10}}}^{\left(-{\mathtt{7}}\right)} = {\mathtt{0.000\: \!000\: \!316\: \!666\: \!666\: \!7}}$$

or $$3\frac{1}{6}*10^{-7}$$

32 is 2⁵ and 243 is 3⁵ so 32/243 is (2/3)⁵.  Is this what you are looking for?

This is actually quite a well-known recreational maths problem.  I first came across it in a Martin Gardner book xxx years ago!  However, in the usual statement of the problem it doesn't tell you the answer.

In the statement of the question here, the hint in the last sentence gives the game away.  In the limit when r is zero, the height, h, becomes the diameter of the sphere.  (In this limit the sphere has an infinitely thin hole of zero volume, so the bead becomes the whole sphere of diameter h).

Bertie is right.

Also, in the line below "Since y(0) = y₀" you have the term -2*0.  This should be +2*0.  However, since the result is zero anyway, this doesn't have any impact on the result!

Here's how it's done for your simple example.  It isn't always possible to get nice results for situations where a(t) and b(t) are very complicated.  For those, use a numerical method!

I need to correct one item above.  In general the integrating factor is $$e^{\int{-a(t)}dt}$$  not $$e^{-a(t)t}$$.

Just noticed Bertie has said this below.

I assume that by "equivalent equation" you mean rearrange to get t = ...

Take log to the base 2 of both sides

log₂y = log₂(2.5*2^t/5)

log₂y = log₂(2.5) + log₂(2^t/5)

log₂y = log₂(2.5) + t/5

t/5 = log₂y - log₂(2.5)

t/5 = log₂(y/2.5)

t = 5log₂(y/2.5)

 5 = f(-3)   ...(1)

 9 = f(8-h)+k   ...(2)

Subtract 4 from both sides of (2)

5 = f(8-h) + k-4   ...(3)

Comparing (1) and (3) it looks like h is 11 and k is 4.

(5, 12) on y = g(x) means 12 = g(5)    ...(1)

(n, 19) on y = g(9x-6)+7 means  19 = g(9n-6)+7   ...(2)

Subtract 7 from both sides of (2) to get 12 = g(9n-6)   ...(3)

Compare (1) and (3).  Clearly 5 = 9n - 6, so n = 11/9

See if you can extrapolate from the following:

You can work it out from the following: