126. Differential Equations
A differential equation specifies a relationship between an unknown function and its derivatives
Solving for (an) unknown function(s) that satisfy this relationship
Initial or boundary conditions to select a unique solution
126.0.1. Linearity
The dependent variable and its derivatives each appear only to the first power, are not multiplied together, and are not inside any other function
Example
Linear
Non-Linear
126.0.2. Homogeneity
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Homogeneous: No terms with just the independent variable of constant
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Nonhomogeneous: Has term(s) with just the independent variable or constants
Example
Homogeneous
Non-Homogeneous
126.0.3. Order & Degree
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Order: highest derivative
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Degree: exponent on highest derivative
Example
Order
Degree
126.0.4. Autonomous
The independent variable does not appear explicitly in the equation (rate of change depends only on the dependent variable)
Example
- Autonomous
- Non-Autonomous
126.0.5. Ordinary Differential Equation (ODE)
- One independent variable
- Involves ordinary derivatives with respect to that variable
126.0.6. Partial Differential Equation (PDE)
- Multiple independent variables
- Involves partial derivatives with respect to those variables
126.0.7. Summary
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Inputs
- Equation itself
- Initial/boundary conditions (optional)
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Outputs
- Function(s) that satisfy it
- Defined for continuous variables
Example
Non-coupled ODE
Exponential Population Growth
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: population at time
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: rate of change of the population at time
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: growth rate constant
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: growth contribution proportional to the current population
Example
Coupled ODE
Lotka-Volterra ODEs (predator-prey)
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= prey population
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= predator population
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: The rate of change of the prey population at time
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: The rate of change of the predator population at time
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: Prey growth rate — how fast the prey multiply in the absence of predators
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: Predation rate — how effectively predators consume prey
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: Predator growth rate — how much predator population increases when consuming prey
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: Predator death rate — how fast predators die without enough food
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: prey naturally reproduce at a rate proportional to how many exist now
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: prey lost to predation, proportional to encounters with predators
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: predators grow in number based on how much prey they consume
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: predators die naturally when there isn’t enough food
126.1. Slope Field (Direction Field)
Example
IVP (Initial Value Problem)
Example
IVP (Initial Value Problem)