Using Mathematica for ODEs, Part 4 (use DSolve, Plot, VectorPlot, Show, Simplify, Solve, & Apart)

First-order scalar equations, part 4. Use DSolve to find the general solution of dy/dt = (1/4)*y*(3-y) (this type of equation arises from the logistic model for population growth) and the unique solution of the initial-value problem dy/dt = (1/4)*y*(3-y), y(0) = 2. Use VectorPlot, Show, and Plot to graph the solution curve on top of the slope field.

(0:00) Introduction
(0:57) Use DSolve to find the general solution of an autonomous equation dy/dt=(1/4)y(3-y)
(4:21) Define a more ideal general solution
(5:23) Use Simplify to confirm the solution
(6:10) Capital E represents the number e=2.71828...
(6:37) Use Solve to solve a generic initial value problem (IVP)
(8:00) How to show the work by hand, including separation of variables
(9:05) Apart will use the method of partial fractions to split a fraction apart
(9:38) Use DSolve for the IVP
(11:18) Use Show, VectorPlot, and Plot to graph the solution in the slope field

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[ChrisUbing]

Hi Bill, my name is Chris Ubing, and I teach physics for a community college in Sierra Vista Az. I have been watching your videos with the idea that I am going to integrate Mathematica into my calc based physics class, specifically the second semester, where they will be dealing with Fields, and various DC and AC circuits. Anyway, I would like to make a couple of comments on these fantastic videos that you shared here.

1. What are your thoughts on defining the equation first i.e.

eqn:= y'[t]==(1/4)y[t]*(3-y[t])

For me, it makes it slightly easier to put eqn into the DSolve command instead of writing out the equation directly into the command.

2. Regarding VectorPlot, I do have a question on why you use a 1 as the first part of the list in the command? Otherwise, thank you very much for demonstrating this command, I've been struggling with it for a while now. Your videos helped a great deal.

Kindest Regards.

Chris Ubing

[lostosecret]

How do I solve when y(t) and that for example y(30); replacing t to variable and get a resulted number?