Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

This equation describes leaking water form a conical tank. We are interested in finding $t$ when $h(t) = 0$ (time it takes to empty the tank).

$$ \frac{dh}{dt} = - \frac{5}{6h^{3/2}}, h(0) = 20 $$

Since this is separable, I separate the equation and solve:

$$ 6h^{3/2}dh = −5dt $$

$$ \frac{12}{5}h^{5/2} = −5t + c $$

Using $h(0) = 20$ I find that $c = 1920\sqrt{5}$. Then, solving $h(t) = 0$, I find that $t= 384\sqrt{5}$.

Why is that the way I distribute the 6/5 have an impact on the solution? All of thesee equations results in different values for $t$:

$$ -6h^{3/2}dh = 5dt $$ $$ \frac{6}{5}h^{3/2}dh = -dt $$ $$ -\frac{6}{5}h^{3/2}dh = dt $$ $$ etc. $$

Why? I understand distributing 6/5 affects $c$, but since it's an arbitrary constant, why does it matter?

Physically, it makes no sense (there can't be an infinite number of times it takes to empty a tank of water).

Thoughts? I must be getting this wrong.

share|improve this question

1 Answer 1

up vote 3 down vote accepted

Are you "solving for $h(t)$" before trying to figure out $h(t)=0$? You can't just plug in $h(t)=0$ and then solve for $t$ in the resulting equation, because $h$ is not independent of $t$. So you must first rewrite $\frac{12}{5}h^{5/2} = -5t + 1920\sqrt{5}$ as $$h(t) = \left(-\frac{25}{12}t + 800\sqrt{5}\right)^{2/5},$$ and then solve $h(t)=0$ for $t$.

Likewise with the other ways of solving the differential equation: before you can solve $h(t)=0$ for $t$, you need to have an explicit expression for $h$ in terms of $t$, rather than an implicit one. Well, you don't have to go all the way to the expression above, it is enough to get to $$(h(t))^{5/2} = -\frac{25}{12}t + 800\sqrt{5}$$

I think you will find that if you clear that constant factor from $h$ first, you will always get the same $t$ for $h(t)=0$.

share|improve this answer
    
Thanks! That clears it up a bit. –  David Chouinard Oct 10 '11 at 11:12

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.