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

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?

Re: DE qn-forming an equaltion with 3 variables

huh,first order DE. i cant seem to integrate it =/

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.?

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.

Re: DE qn-forming an equaltion with 3 variables

Yes, exactly! Have you completed the partial fraction decomposition?

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.

Re: DE qn-forming an equaltion with 3 variables

So what did you get for B and F?

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.

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?

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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.

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.

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)].

Re: DE qn-forming an equaltion with 3 variables

I didnt know it was integration by partial fractions,didnt realise! Had me thinking for days! thanks guys!!

Re: DE qn-forming an equaltion with 3 variables

Your partial fraction decomposition using the cover-up method is correct, however, your anti-derivative is not. It appears you switched B and F.