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

First-Order Differential Equations

What Makes It 'First-Order'?

A first-order differential equation involves only the first derivative of the function. These are the simplest kind and a great place to start!

Separable Equations

Some equations can be rewritten so all the y's are on one side and all the x's are on the other. This makes them easy to solve by integrating both sides.

Linear First-Order Equations

These look like \( \frac{dy}{dx} + P(x)y = Q(x) \). There's a special trick called the integrating factor to help solve them.

Why They're Useful

Many natural processes, like radioactive decay and population growth, are modeled by first-order equations.

Examples

  • Solving \( \frac{dy}{dx} = 3y \) gives \( y = Ce^{3x} \), describing exponential growth.

  • For \( \frac{dy}{dt} + 2y = 4 \), we use an integrating factor to find the solution.

In a Nutshell

First-order differential equations model simple changing systems and are often solvable using integration.

Next
Second-Order Differential Equations