HINT : It's an Exact Equation...
Mathstud's Limit Marathon has inspired me to start a Differential Equation Marathon. It will have similar rules : "after someone completes a DE and gets the OK from the DE's poster, he/she posts their own DE."
There are no restrictions on how you may solve them. But please show all the steps that led you to your solution.
Here's an easy one:
Solve
That's good! However, I was taught that the solution should have the form , so my answer was .
Word problems are fine. They can be challenging (depending on how you interpret the question to write your DE); hence, that's the reason why I like them. You can post your question now, Galactus.
Here is a decent one. I like problems like this (it's not that bad):
The conical tank in the drawing loses water out of the hole in the bottom.
The cross-sectional area of the hole is 1/4 ft^2. Determine a DE representing
the height of the water h at ant time. Ignore friction and contraction of the
water stream at the hole.
Hint: you can use
to
represent the height at any time, but use the given data.
Torricelli's law tells us that the speed of the water which flows out of the tank is . (the hole is small enough so that this is correct) As there's no "contraction of the water stream at the hole" (conservation of volumetric flow rate)
Substituting gives us
which can be rewritten
Integrating :
Using the graph, one can see that the function is linear for which could be interesting to make an hourglass. (it'd be a large one )