Separable Equations  MIT 18.03SC Differential Equations, Fall 2011  Summary and Q&A
TL;DR
This video explains how to solve separable equations and find both specific and general solutions using the separation of variables method.
Key Insights
 🙃 Separable equations can be solved by regrouping variables and integrating both sides.
 😚 Imposing conditions ensures that the solution satisfies the initial value problem and recovers lost solutions.
 🚦 Solutions of separable equations may exhibit complex behavior, such as vertical asymptotes.
 ❓ The general solution includes all possible solutions, including those that may be excluded due to imposed conditions.
Transcript
PROFESSOR: Welcome to this session on separable equations. So in this problem, you're asked in the first question to solve the initial value problem dy/dx equals y square with the initial condition y of zero equals 1. In the second part of the problem, you're asked to find the general solution where no initial condition is imposed. So here you need... Read More
Questions & Answers
Q: How do you solve a separable equation?
To solve a separable equation, you regroup the variables and integrate both sides, usually resulting in an expression with an undetermined constant of integration.
Q: Why is it important to impose conditions in the solution of the problem?
Imposing conditions in the solution, such as y not equal to zero, helps recover lost solutions and ensures that the solution satisfies the initial value problem.
Q: What does it mean for a solution to have a vertical asymptote?
A vertical asymptote occurs at a value of x for which the solution becomes undefined or goes to infinity. In this case, the solution y(x) = 1/(1x) has a vertical asymptote at x = 1.
Q: What is the general solution of the separable equation?
The general solution includes all possible solutions, including the lost solution. In this case, the general solution includes y(x) = 1/(1x) (excluding y = 0) and y(x) = 0.
Summary & Key Takeaways

The video teaches how to solve a separable equation dy/dx = y^2 with the initial condition y(0) = 1.

The separation of variables method is used to regroup the variables and integrate both sides of the equation.

The solution, y(x) = 1/(1x), has a vertical asymptote at x = 1 and consists of two parts: one that goes to infinity and one that goes to zero.