Joseph M Mahaffy h jmahaffysdsuedu i Lecture Notes Separable Differential

Joseph m mahaffy h jmahaffysdsuedu i lecture notes

This preview shows page 6 - 14 out of 41 pages.

Joseph M. Mahaffy, h [email protected] i Lecture Notes – Separable Differential Equation — (6/41)
Image of page 6

Subscribe to view the full document.

Introduction Separation of Variables Modified Malthusian Growth Model Examples General Separable Differential Equation Example of Desiccation of a Cell 1 Desiccation of a Cell: The model satisfies dV dt = - kV 2 / 3 Suppose that the initial volume of water in the cell is V (0) = 8 mm 3 Suppose that 6 hours later the volume of water has decreased to V (6) = 1 mm 3 Solve this differential equation Find k and graph the solution Determine when all of the water has left the cell Joseph M. Mahaffy, h [email protected] i Lecture Notes – Separable Differential Equation — (7/41)
Image of page 7
Introduction Separation of Variables Modified Malthusian Growth Model Examples General Separable Differential Equation Example of Desiccation of a Cell 2 Solution: The model is a separable differential equation dV dt = - kV 2 / 3 Separate variables to give Z V - 2 / 3 dV = - Z k dt Upon integration, 3 V 1 / 3 ( t ) = - kt + C Equivalently, V ( t ) = - kt + C 3 3 The initial condition gives V (0) = 8 = ( C 3 ) 3 or C = 6 Joseph M. Mahaffy, h [email protected] i Lecture Notes – Separable Differential Equation — (8/41)
Image of page 8

Subscribe to view the full document.

Introduction Separation of Variables Modified Malthusian Growth Model Examples General Separable Differential Equation Example of Desiccation of a Cell 3 Solution: The model is given by V ( t ) = - kt + 6 3 3 The other condition gives V (6) = 1 = - 6 k + 6 3 3 = ( - 2 k + 2) 3 So k = 1 2 The solution to this problem is V ( t ) = 2 - t 6 3 The solution vanishes (all the water evaporates) at t = 12 Joseph M. Mahaffy, h [email protected] i Lecture Notes – Separable Differential Equation — (9/41)
Image of page 9
Introduction Separation of Variables Modified Malthusian Growth Model Examples General Separable Differential Equation Example of Desiccation of a Cell 4 Graphs of Desiccation of a Cell 0 2 4 6 8 10 12 0 2 4 6 8 Time (hr) Volume ( μ m 3 ) Dessication of a Cell Joseph M. Mahaffy, h [email protected] i Lecture Notes – Separable Differential Equation — (10/41)
Image of page 10

Subscribe to view the full document.

Introduction Separation of Variables Modified Malthusian Growth Model Examples General Separable Differential Equation Example 1 - Separable Differential Equation 1 Example - Separable Differential Equation Consider the differential equation dy dt = 2 ty 2 Solution: Separate the variables t and y Put only 2 t and dt on the right hand side And only y 2 and dy are on the left hand side The integral form is Z dy y 2 = Z 2 t dt Joseph M. Mahaffy, h [email protected] i Lecture Notes – Separable Differential Equation — (11/41)
Image of page 11
Introduction Separation of Variables Modified Malthusian Growth Model Examples General Separable Differential Equation Example 1 - Separable Differential Equation 2 Solution (cont) The two integrals are Z dy y 2 = Z 2 t dt The two integrals are easily solved - 1 y = t 2 + C Note that you only need to put one arbitrary constant , despite solving two integrals This is easily rearranged to give the solution in explicit form y ( t ) = - 1 t 2 + C Joseph M. Mahaffy, h [email protected] i Lecture Notes – Separable Differential Equation — (12/41)
Image of page 12

Subscribe to view the full document.

Introduction Separation of Variables
Image of page 13
Image of page 14
  • Fall '08
  • staff

What students are saying

  • Left Quote Icon

    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.

    Student Picture

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

  • Left Quote Icon

    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.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    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.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern

Ask Expert Tutors You can ask You can ask ( soon) You can ask (will expire )
Answers in as fast as 15 minutes