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...
Basic Concepts

Second-Order Differential Equations

Going Further: Second Derivatives

Sometimes, the rate of change of the rate of change matters! Second-order differential equations involve up to the second derivative.

Common Appearances

These equations show up in physics, especially whenever you have things like acceleration (the second derivative of position).

Homogeneous vs. Nonhomogeneous

  • Homogeneous: All terms involve the function or its derivatives.
  • Nonhomogeneous: An extra term (not involving the function) is present.

Solving Methods

Characteristic equations and undetermined coefficients are common tools to crack these puzzles.

Examples

  • The equation \( m\frac{d^2x}{dt^2} + kx = 0 \) models a mass on a spring.

  • Electrical circuits with inductors and capacitors use second-order ODEs.

In a Nutshell

Second-order equations help us model motion, vibrations, and many more complex systems.

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Systems of Differential Equations