Problem_1.7soln

# Problem_1.7soln - First Order ODEs Problem_1.7 The growth...

• Homework Help
• 2

This preview shows page 1 - 2 out of 2 pages.

First Order ODEs Problem_1.7 The growth of a biological population P(t) is given by the Verhulst Pearl equation (1) P ÅÅÅÅÅÅÅ t = kP J 1 - P ÅÅÅÅÅÅÅ P N where P is the limiting size of the population, beyond which the population growth is zero. If P 0 is the initial population at t=0, find an expression for the population at a later time t. (Hint: use partial fractions where necessary) Solution First we rearrange the equation as follows (2) P ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅ P - P 2 ê P = k t Using partial fractions we can write (3) 1 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅ P - P 2 ê P = 1 ÅÅÅÅ P + 1 ê P ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅ 1 - P ê P Using the partial fraction representation and integrating once gives (4) 1 ÅÅÅÅ P P + 1 ÅÅÅÅÅÅÅ P 1 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅ 1 - P ê P P = k t The result is (5) Ln H P L - Ln H 1 - P ê P L = kt + C

Subscribe to view the full document.

• Summer '13
• LeThuy
• Elementary algebra, DSOLVE, P¶, Ø ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

{[ snackBarMessage ]}

###### "Before using Course Hero my grade was at 78%. By the end of the semester my grade was at 90%. I could not have done it without all the class material I found."
— Christopher R., University of Rhode Island '15, Course Hero Intern

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern