# DE qn-forming an equaltion with 3 variables

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• Nov 6th 2012, 07:57 PM
DE qn-forming an equaltion with 3 variables
the growth of an insect is studied.The rate of death is proportional to the number of insects,x, (in thousands)at any time, t days after the start of the experiment.rate of birth is proportional to x square.When x=2,no. of larvae hatched equals no. of insects that died. show dx/dt = ax(x-2), where a is a constant. Given that x=a initially,find an equation relating x,a and t where x,a>2.

I can do the show part,but cant figure out the eqn.:S
• Nov 6th 2012, 09:04 PM
MarkFL
Re: DE qn-forming an equaltion with 3 variables
You are being asked to solve the IVP:

$\displaystyle \frac{dx}{dt}=ax(x-2)$ where $\displaystyle x(0)=a$

What kind of ODE is this?
• Nov 6th 2012, 09:09 PM
Re: DE qn-forming an equaltion with 3 variables
huh,first order DE. i cant seem to integrate it =/
• Nov 6th 2012, 09:14 PM
MarkFL
Re: DE qn-forming an equaltion with 3 variables
Yes, it is first order, but what type of first order ODE, as in linear, separable, exact, etc.?
• Nov 6th 2012, 09:21 PM
Cbarker1
Re: DE qn-forming an equaltion with 3 variables
The ode is a separable equation by the x terms; Integrate by partial fractions. Solve for x in terms of t.
• Nov 6th 2012, 09:24 PM
MarkFL
Re: DE qn-forming an equaltion with 3 variables
Yes, exactly! Have you completed the partial fraction decomposition?
• Nov 6th 2012, 09:37 PM
Cbarker1
Re: DE qn-forming an equaltion with 3 variables
yes, I had done it in the past.
Set up: B/(x-2)+F/x=1/x(x-2)
2. Multiply the x(x-2) both sides: B/(x-2)+F/x=1/x(x-2)
3. plug in values of x to solve the B and F in the equation: Bx+F(x-2)=1
4. Use the values to finish the integration. Before you solve the equation in terms of t. Plug-in the intial value of the problem and solve for the constant. Then solve the equation in terms of t or let it be in implicit form.
• Nov 6th 2012, 09:41 PM
MarkFL
Re: DE qn-forming an equaltion with 3 variables
So what did you get for B and F?
• Nov 6th 2012, 09:47 PM
Cbarker1
Re: DE qn-forming an equaltion with 3 variables
they are just constants variable for the partial fractions. I need to use different variables to what solve the constant variable; I could pick any letter of the alphabet if i desire.
• Nov 6th 2012, 09:57 PM
MarkFL
Re: DE qn-forming an equaltion with 3 variables
Yes, you may use any character, even those not in the alphabet, such as a smiley face or a heart, but what I mean is what numeric value did you obtain for them? How did you rewrite the expression to be integrated using partial fractions?
• Nov 6th 2012, 10:07 PM
Cbarker1
Re: DE qn-forming an equaltion with 3 variables
If you are still confused with how I am directing you to complete the problem, use the attachment. To see how the partial fraction method works.
• Nov 6th 2012, 10:14 PM
MarkFL
Re: DE qn-forming an equaltion with 3 variables
I know how the method works, in fact I would use the Heaviside cover-up method instead. I am trying to find why you are unable to integrate.
• Nov 6th 2012, 10:18 PM
Cbarker1
Re: DE qn-forming an equaltion with 3 variables
I can integrate the integrand. Pretty easily. The solution is the 1/2[ln(x)-ln(x-2)].
• Nov 6th 2012, 10:20 PM