📂Odinary Differential Equations

Definition and Examples of Differential Equations

Definition

A differential equation is an equation that includes derivatives of one or more dependent variables with respect to one or more independent variables.

$$\dfrac{dy}{dx}=y$$

$$\dfrac{d^2y}{dx^2} = y$$

Explanation

Most physical situations can be described by first-order or second-order differential equations.

Falling Body

$$F=ma=mg$$

$$v=\dfrac{dy}{dt}$$

$$a=\dfrac{dv}{dt}=\dfrac{d}{dt} \left( \dfrac{dy}{dt} \right)=\dfrac{d^2y}{dt^2}$$

$$\dfrac{d^2y}{dt^2}=g$$

Spring Mass System

$$F=ma=-ky$$

$$a= -\dfrac{k}{m}y$$

$$\dfrac{d^2y}{dt^2}=-\dfrac{k}{m}y$$

$$\dfrac{d^2y}{dt^2}+\dfrac{k}{m}y=0$$

When this is referred to as $w^2=\dfrac{k}{m}$, then,

$$\dfrac{d^2y}{dt^2}+w^2y=0$$

RLC Circuit

$$L\dfrac{d^2q}{dt^2}+R\dfrac{dq}{dt}+\dfrac{1}{c}q=V(t)$$

Schroedinger Equation

$$i\hbar \dfrac{\partial \psi}{\partial t}=-\dfrac{\hbar^2 }{2m} \dfrac{\partial^2 \psi}{\partial x^2} + u(x)\psi$$