Differential Equations

Study of equations involving derivatives and their applications.

What Are Differential Equations?First-Order Differential EquationsSecond-Order Differential Equations
Systems of Differential EquationsPartial Differential EquationsNonlinear Differential Equations and Chaos
Modeling Population GrowthEngineering Vibrations and OscillationsMedical and Biological Applications
How to Approach Differential Equation ProblemsTips for Test Day Success
What Are Differentia...First-Order Differen...Second-Order Differe...Systems of Different...Partial Differential...Nonlinear Differenti...Modeling Population ...Engineering Vibratio...Medical and Biologic...How to Approach Diff...Tips for Test Day Su...
Advanced Topics

Systems of Differential Equations

When One Equation Isn't Enough

Sometimes, processes are interconnected. Systems of differential equations let us study multiple changing quantities that influence each other.

Matrix Methods

For linear systems, you can use matrices and vectors to organize and solve the equations—no need to panic, it's just a tidy way to keep track!

Real-World Use

Predator-prey models (like rabbits and foxes) are classic examples of systems.

Solving Strategies

  • Use substitution or elimination for small systems.
  • For bigger systems, try eigenvalues and eigenvectors.

Key Formula

\( \vec{x}' = A\vec{x} \)

Examples

  • Lotka-Volterra equations model how predator and prey populations change together.

  • Modeling chemical reactions with several interacting substances.

In a Nutshell

Systems allow us to model and analyze multiple changing things at once, especially when they interact.

Key Terms

Eigenvalue
A special number associated with a matrix that tells you about the system's behavior.
Vector
An object with both magnitude and direction, used to represent many variables at once.
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Partial Differential Equations