10 Examples of Differential Equations

Differential equations are a fundamental concept in mathematics and science, used to describe how quantities change in relation to one another. They play a important role in various fields, from physics to engineering and biology.

In this article, we will explore examples of differential equations.

Examples of Differential Equations

These are 10 examples of differential equations.

1: First-Order Linear Differential Equation

A first-order linear differential equation is one of the simplest types and is used to describe a rate of change that is directly proportional to the current state.

For example, dy/dx = y

2: Second-Order Linear Differential Equation

This equation involves the second derivative and is commonly used to describe physical systems with acceleration and force, such as a mass on a spring.

For example,

y'' + g = 0  

3: Simple Exponential Growth

Differential equations can describe exponential growth, where a quantity’s rate of change is proportional to its current value. For example,

dy/dt = ky

4: Newton’s Law of Cooling

This equation describes how the temperature of an object changes as it cools or heats up in a surrounding environment. For example

dT/dt = -k(T - T_s)

5: Harmonic Oscillator Equation

Found in physics, this equation describes the motion of a particle subjected to a restoring force. For example,

m̈ + kx = 0

6: Euler’s Differential Equation

Euler’s equation is a famous example in mathematics, often used in solving problems involving complex numbers. Fir example

ay” + by’ + cy = 0

7: Logistic Growth Equation

This equation models population growth when resources are limited, and growth rates slow as the population approaches a carrying capacity. For example,

dy/dt = ky(1 – y/K)

8: Pendulum Differential Equation

It describes the motion of a pendulum, which swings back and forth under the influence of gravity. For example,

θ” + (g/L)sin(θ) = 0

9: Wave Equation

Used extensively in physics and engineering, the wave equation describes the behavior of waves, such as sound or light. For example,

∂²u/∂t² = v²∇²u

10: Schrödinger Equation

In quantum mechanics, the Schrödinger equation is fundamental, describing how the wave function of a physical system changes over time. For example,

iℏ∂ψ/∂t = −(ℏ²/2m)∇²ψ + Vψ