A Differential Equation is an equation that involves derivatives of functions with respect to independent variables.
Outlined below are a few examples:
Derivatives can be written using different notations summarized below:
| Notation | Examples |
| Leibniz | \( \cfrac{dy}{dx}, \cfrac{d^3u}{dv^3} \) |
| Prime | \(y^{'}, u^{''}, v^{(4)}\) |
| Dot | \(\dot{x}, \ddot{y} \) |
| Subscript | \(\cfrac{\partial^2y}{\partial x \partial t} = \cfrac{\partial}{\partial x} \left(\cfrac{\partial y}{\partial t}\right) = y_{tx}\) |
| Ordinary Differential Equations (ODEs) contain only ordinary derivaties with respect to a single independent variable. | \(\cfrac{dy}{dx} + 3y - \cos t = 0\) \( \ddot x - 2\dot x = e^t\) |
| Partial Differential Equations (PDEs) contain partial derivaties with respect to several independent variable. | \(\cfrac{du}{dx} - 3\cfrac{du}{dy} = 0 \) \( v_x \sin t + v_y \cos t = v_{tt} \) |
The order of a differential equation is the order of the highest derivative in the equation. The degree of a differential equation is the degree of the highest term.
Determine the order and degree of \( \ddot x + \left(\cfrac{dx}{dt}\right)^3 = x \tan t \).
The order is the highest derivative which is \(2\) on the term \(\ddot x\). The degree is the largest exponent which is on \(3\) on term \(\left(\cfrac{dx}{dt}\right)^3\).
A differential equation is linear if it is linear in terms of \(y, y^{'}, \dots y^{(n)} \). These terms can have coefficients with non-linear terms of the independent variable. The general form of a linear DE is:
\( a_n(x) \cfrac{d^ny}{dx^n} + a_{n-1}(x) \cfrac{d^{n-1}y}{dx^{n-1}} + \dots + a_1(x) \cfrac{dy}{dx} + a_0(x) y = f(x) \)
Determine whether the DE is linear \( 2xy^{'''} -5x^2y^{'} = e^{2x}\).
The equation is linear. Although there is an \(x^2\), it is non-linear with respect to the independent variable so it doesn't impact the linearity. Same goes for \(e^{2x}\).