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<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> | <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> | ||
| + | |||
| + | '''·''' When <math>a=0</math>, <math>b≠0</math>, there is no solution. | ||
| + | |||
| + | '''·''' 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. | ||
Revision as of 16:41, 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 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.
