Homework Help: Differential Equation Problem

1. Oct 20, 2015

Charge2

1. The problem statement, all variables and given/known data
This is a interesting (morbid) problem from Simmons- Calculus with Analytic Geometry.
In a certain barbourous land, two neighbouring tribes have hated one another from time immemorial. Being barbourous peoples, their powers of belief are strong, and a solemn curse pronounced by the medicine man of the first tribe deranges and drives them to murder and suicide. If the rate of change of the population P of the second tribe is $-\sqrt{P}$ per week, and if the population is 676 when the curse is uttered, when will they all be dead?

Intial Conditions
$P(0) = 676$

2. Relevant equations

None

3. The attempt at a solution

$\frac{dP}{dt} = -\sqrt{t} = -t^{1/2}$.

Separating the differential equation,

$dP = -t^{1/2}dt$,

Then, by intergrating,

$\int dP = - \int t^{1/2}dt$

$P = -\frac{2}{3} t^{3/2}+ C$......(1)

Solving for C at t= (0) or P(0) weeks, when the medicine man uttered his curse,

$676 = -\frac{2}{3} (0)^{3/2} + C$,
$C = 676$.

Subbing this in (1)

$P = -\frac{2}{3} t^{3/2}+ 676$ ......(2)

Rearanging (2) for t, when P = 0 because the second tribe are all dead,

$0 = -\frac{2}{3} t^{3/2}+ 676$,
$-676 = -\frac{2}{3} t^{3/2}$,
$t = (\frac{2028}{2})^{2/3}= 100.93 = 101$ weeks .

Is this correct. Or have I made a massive error? Seems like they need a more powerful medicine man...

Last edited: Oct 20, 2015
2. Oct 20, 2015

SteamKing

Staff Emeritus
Yes, you have made a massive error.

According to the OP, "the rate of change of the population P of the second tribe is $-\sqrt{P}$ per week", yet you have set your ODE = $-\sqrt{t}$. Why is that?

3. Oct 20, 2015

hunt_mat

No, the equation you need is $$\frac{dP}{dt}=-\sqrt{P}$$

4. Oct 20, 2015

Charge2

Dang. I had this on my first attempt but it just looked wrong and unfamiliar, so played around with the equation, this was the first attempt,

$\frac{dP}{dt} = -\sqrt{P} = -P^{1/2}$.
and rearanged it to,

$t = \int \frac{1}{-p^{1/2}}dP =$.

Is that ok?

5. Oct 20, 2015

SteamKing

Staff Emeritus

6. Oct 20, 2015

Charge2

Ok this is not working out,
$t = -2\sqrt{P} + C$
C = 52
$t = -2\sqrt{P} + 52$
$t = 0.$

7. Oct 20, 2015

PWiz

This is the correct solution. Substitute P=0 to find t.

8. Oct 20, 2015

Charge2

52 weeks... not a bad medicine man after all. I on the other hand, need to work more on ode magick.