| Line 24: | Line 24: | ||
'''<big><big><big> 2.1 Concept <big><big><big>''' | '''<big><big><big> 2.1 Concept <big><big><big>''' | ||
| − | <font size="3px">From the example from 1.1, here we still suppose that we had a linear equation <math>ax+b=0</math> with respect to <math>x</math>. | + | <font size="3px">From the first example from 1.1, here we still suppose that we had a linear equation <math>ax+b=0</math> with respect to <math>x</math>. |
'''·''' When <math>a=0</math>, <math>b≠0</math>, there is no solution. | '''·''' When <math>a=0</math>, <math>b≠0</math>, there is no solution. | ||
| Line 30: | Line 30: | ||
'''·''' When <math>a≠0</math>, there is one solution <math>x=-\frac{b}{a}</math>. | '''·''' When <math>a≠0</math>, there is one solution <math>x=-\frac{b}{a}</math>. | ||
| − | '''·''' When <math>a=b=0</math>, there are infinitely many solutions to this linear equation. </font> | + | '''·''' When <math>a=b=0</math>, there are infinitely many solutions to this linear equation. |
| + | |||
| + | Similarly, an ODE may also have no solution, a unique solution or infinitely many solutions.</font> | ||
Revision as of 17:00, 16 October 2017
The Existence and Uniqueness Theorem for Solutions to ODEs
2.0 Abstract
Before starting this tutorial, you are supposed to be able to:
· Find an explicit solution for $ \frac{dy}{dt}=f(t) $. This is the same thing as finding the integral of $ f(t) $ with respect to $ t $.
· Know the difference between general solution and a solution satisfying the initial conditions.
· Check one function is a solution to an ODE.
· Distinguish ODE and PDE, know the usual notations.
· Know the basic concepts of ODEs (order, linearity, homogeneity, etc).
2.1 Concept
From the first example from 1.1, here we still suppose that we had a linear equation $ ax+b=0 $ with respect to $ x $.
· When $ a=0 $, $ b≠0 $, there is no solution.
· When $ a≠0 $, there is one solution $ x=-\frac{b}{a} $.
· When $ a=b=0 $, there are infinitely many solutions to this linear equation.
Similarly, an ODE may also have no solution, a unique solution or infinitely many solutions.
