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Solving Differential Equations
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An introduction to differential equations and how to solve them.
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At this point, we've extensively studied the derivatives of functions f′(x) to analyze how a known quantity changes or evolves. For example, in kinematics we learned to find the velocity and acceleration knowing only the position x(t). When we know the rules for how something behaves, and we can model it as a function, we have all these tools for analyzing how it changes.
But in the real world, we often don't know have a function that tells us what the behavior is. In fact, most physical laws of the universe only tell us how things change, not how things actually are at any given time.
A classic example is water leaving a tank.
Since the pressure at the bottom of the tank is proportional to the height H of water in the tank, the water leaves faster when H is bigger. So as the tank empties, the rate at which the water leaves decreases.
Separable Variables
AHL AI 5.14
When you have a differential equation in the form
dxdy=f(x)g(y)
you can bring all the y's to one side and all the x's to the other:
g(y)1dy=f(x)dx
And then integrate:
∫g(y)1dy=∫f(x)dx
Example:
dxdy=yx2
∫ydy=∫x2dx
2y2=3x3+C
y=±√32x3+2C
Particular Solutions
AHL AI 5.14
The solutions to differential equations will usually contain a constant of integration +C. These are called general solutions.
Often, we are given an initial condition, ie the value of y for a specific x, which we can use to solve for C. The result is the particular solution.
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