How do you solve the Initial value probelm dpdt=10p(1−p), p(0)=0.1 Solve and show that p(t)→1 as t→∞.

Name: How do you solve the Initial value probelm dpdt=10p(1−p), p(0)=0.1 Solve and show that p(t)→1 as t→∞.
Uploaded: 2022-02-23T17:22:57+00:00
Description: How do you solve the Initial value probelm dpdt=10p(1−p), p(0)=0.1 Solve and show that p(t)→1 as t→∞.

Question

How do you solve the Initial value probelm  dpdt=10p(1−p),  p(0)=0.1  Solve and show that p(t)→1 as t→∞.

Jenny Sheppard · Accepted Answer

This is the so-called logistic equation, which occurs often in population dynamics and many other contexts. There&#039;s a trick which works for this particular equation and is much simpler than separation of variables (in my opinion): change variables to y(t)=1p(t). Then the nonlinear equation for p turns into an inhomogeneous linear equation for y, which can be solved immediately by the usual &#039;&#039;homogeneous+particular solution&#039;&#039; method (the homogeneous solution is an exponential, and the particular solution is a constant). Since this is tagged as homework, I&#039;ll let you have a go at the details yourself.

Philip Williams · Accepted Answer

The method we can use here is called Separation of Variables. Take all the ps

alenahelenash · Accepted Answer

From your comment, it looks you have been able to integrate correctly, following Ragib&#039;s Hint and Gourtaur comment. But now your problem is (to finish the solution) to express p(t). This rest part is a simple algebra. Let me express p(t) in terms of t:p1−p=e10t+10c=e10t⋅e10c=k.e10t (where k=e10c is a new constant)⇒p(1−p)+p=ke10t1+ke10t (I applied ab=cd⇒ab+a=cd+c. You can just multiply both sides by (1−p), or cross-multiply and solve for p)⇒p=p(t)=11+k′e−10t (dividing numerator and denominator of the fraction on RHS by ke10t and writing ke10t and writing k′=1k)Now, from using the condition p(0)=0.1=110, we get 110=11+k′⇒k′=9

How do you solve the Initial value probelm \frac{dp}{dt}=10p(1-p), p(0)=0.1 Solve and show

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