# 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 Growt**h

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

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