A first order equation in mathematics can refer to a first order differential equation. According to the reference, a first order differential equation is an equation of the form F(t,y,˙y)=0.
Understanding First Order Differential Equations
Let's break this down:
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F(t,y,˙y)=0: This is the general form. It means we have some function, 'F', that involves:
- 't': An independent variable (often representing time).
- 'y': A dependent variable (a function of 't', i.e., y = f(t)).
- '˙y': The first derivative of 'y' with respect to 't' (i.e., dy/dt or f'(t)). This is where the "first order" part comes from – only the first derivative is involved.
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Solution: A solution to this equation is a function f(t) which satisfies the original equation when plugged in. In other words, F(t,f(t),f′(t))=0 for every value of t.
Example
Imagine a simple first-order differential equation:
˙y = -2y
This states that the rate of change of y (˙y) is proportional to y itself. A possible solution could be y = e-2t. If we substitute this into the equation, we find that it holds true.
