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Solving Differential Equations
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Solving differential equations by direct integration, separable variables, homogeneous equations using the substitution y=vx, and linear first-order equations with an integrating factor, then using initial conditions to find particular solutions.
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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.
Direct Integration
AHL 5.18
The easiest differential equations to solve are the ones in the form
dxdy=f(x)
as we can simply integrate:
y=∫f(x)dx🚫
Separable Variables
AHL 5.18
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 5.18
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.
Integrating Factor
AHL 5.18
For a differential equation in the form
dxdy+P(x)y=Q(x)
Multiply both sides by integrating factor (often called μ):
e∫P(x)dx📖
and notice the product rule on the LHS.
Homogeneous Equation
AHL 5.18
dxdy=f(xy)🚫
Let y=vx, then v=xy and dxdy=v+x⋅dxdv.
Note: On IB exams you will be told to use the substitution y=vx.
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