Joseph M Mahaffy h jmahaffysdsuedu i Lecture Notes Separable Differential

# Joseph m mahaffy h jmahaffysdsuedu i lecture notes

• 41

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

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

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)
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)

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)
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)

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)
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)

Subscribe to view the full document.

Introduction Separation of Variables
• Fall '08
• staff

### 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