differential equation problem

**swimmergirl** · March 20th 2009, 04:25 AM

I almost have the answer but I can't quite get there.

Differential equation is

dH/dt= (40-H)^2/100

I solved it through here:

H= 40 - 200/(2t + 5)
However, when I plug in a value t=5 I get the wrong answer. My professor has this but the following step says:

H=80t/(2t+5)

How do I get to that?

**mr fantastic** · March 20th 2009, 04:29 AM

Originally Posted by swimmergirl

I almost have the answer but I can't quite get there.

Differential equation is

dH/dt= (40-H)^2/100

I solved it through here:

H= 40 - 200/(2t + 5)
However, when I plug in a value t=5 I get the wrong answer. My professor has this but the following step says:

H=80t/(2t+5)

How do I get to that?

$\frac{dt}{dH} = \frac{100}{(40 - H)^2}$

$\Rightarrow t = \int \frac{100}{(40 - H)^2} \, dH = \frac{100}{40 - H} + C$

where C is a constant whose value cannot be calculated without additional data (such as H = 0 at t = 0 ....?)

**swimmergirl** · March 20th 2009, 04:44 AM

correct...initial conditions are H(0)=0. Sorry these weren't given in the problem but its about the height of a tree over time. Therefore, I just put that initially it isn't growing which suffices H(0)=0 and I forgot to mention that.

**mr fantastic** · March 20th 2009, 05:10 PM

Originally Posted by swimmergirl

correct...initial conditions are H(0)=0. Sorry these weren't given in the problem but its about the height of a tree over time. Therefore, I just put that initially it isn't growing which suffices H(0)=0 and I forgot to mention that.

$t = \frac{100}{40 - H} + C$

Substitute $H = 0, ~ t = 0$ : $0 = \frac{100}{40} + C \Rightarrow C = - \frac{5}{2}$ .

Therefore:

$t = \frac{100}{40 - H} - \frac{5}{2} \Rightarrow 2t + 5 = \frac{200}{40 - H}$

$\Rightarrow \frac{1}{2t + 5} = \frac{40 - H}{200} \Rightarrow \frac{200}{2t + 5} = 40 - H$

$\Rightarrow H = 40 - \frac{200}{2t + 5} = \, ....$

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