Differential Equations
Differential equations are crucial for the study of many physical phenomena and dynamical systems. In engineering, differential equations describe various mechanical, electrical, hydraulic, and thermal systems.
A differential equation is an equation that is composed of a single unknown function along with its derivates. An example of a simple differential equation is
$$ \dot{y} = f(x) $$Another example with the second derivative of the unknown function appearing in the differential equation is as follows
$$ \ddot{y} + c^2 y = \cos(x) $$Dot Notation
In this article, the derivative is represented using the flyspeck or dot notation, see Newton's notation. For example, the second derivative is represented as
$$ \ddot{y} \equiv \frac{d^2y}{dx^2} = y''_x $$Whereas an nth-derivative is written as
$$\overset{n}{\dot{y}}\equiv \frac{d^ny}{dx^n} = y^{(n)}_x$$Ordinary vs Partial DEs
Ordinary DEs (ODEs)
If the unknown function of the differential equation has only a single independent variable, it is called an ordinary differential equation. Since the unknown function $y$ in the following differential equation has only a single independent variable $x$, it is an ordinary differential equation.
$$ \ddot{y} + c^2 y =\frac{d^2y}{dx^2} + c^2 y=\cos(x) $$General Form of Nth-Order Ordinary Differential Equation
The general form of an Nth-order ordinary differential equation is expressed as follows:
$$ F\left(x,y,\dot{y},\ddot{y},\dots,\overset{n-1}{\dot{y}},\overset{n}{\dot{y}}\right) = 0 $$Where $x$ is the independent variable and $y$ is the unknown function.
Partial DEs (PDEs)
On the other hand, when the unknown function of the differential equation depends on two or more independent variables, it is called a partial differential equation. In the following example, the unknown function $u$ depends on both $x$ and $y$, which are independent variables.
$$ u_{xx} + u_{yx} = \frac{\partial^2u}{\partial x^2} + \frac{\partial^2 u}{\partial x\partial y}= \frac{\partial^2u}{\partial x^2} + \frac{\partial}{\partial x}\left(\frac{\partial u}{\partial y}\right)= 0 $$Order of Differential Equation
The order of a differential equation is determined by the order of the highest derivative of the unknown function. For example, the following differential equation is a second-order differential equation,
$$\ddot{y} + \sin(y)=0$$Whereas the following equation is a third-order differential equation,
$$\dddot{y} + x^2 \ddot{y} = e^{2x}$$Linearity
There is a distinction between linear and nonlinear differential equations.
Linear ODEs
Linear Nth-Order (Ordinary) Differential Equation
A linear nth-order (ordinary) differential equation is expressed as follows:
$$ \begin{equation} \label{eq:lode} f_n(x)\overset{n}{\dot{y}} + f_{n-1}(x) \overset{n-1}{\dot{y}} + \dots + f_1(x) \dot{y} + f_0(x) y = F(x) \end{equation} $$Where $f_0(x), f_1(x), \dots, f_n(x)$ are known as the coefficients of the differential equation.
The coefficients of a linear ODE could be either functions of the independent variable or constants.
Examples of linear ODEs
Two examples of linear 2nd-order ODEs are written as follows:
$$\ddot{y} + 30 \dot{y} + 114 y = \sin(x)$$and
$$62\sqrt{x} \ddot{y} + 30x^2 \dot{y} + 114 x y = 0$$Nonlinear ODEs
On the other hand, an ODE is nonlinear if it can't be expressed by the formula $\eqref{eq:lode}$.
Examples of nonlinear ODEs
Two examples of nonlinear ODEs are expressed as follows:
$$\ddot{y} + 30 \dot{y}^2 + \cos(y) = 10$$and
$$\ddot{y}\dot{y} + \sin(x)\cot(\dot{y}^2) + y = 0$$As can be seen, the function $x$ and/or its derivatives are involved in arithmetic and trigonometric operations.