Basics of Differential Equation
What is Differential Equation?
Differential Equation is an equation that has a function (say y), and one or more of its derivatives ($\frac{dy}{dx}$).
For example, y + $\frac{dy}{dx}$ = 3x
Order and Degree of Differential Equations
Order - The highest derivative in the Differential Equation, i.e. first derivative ($\frac{dy}{dx}$), second derivative ($\frac{d^2y}{dx^2}$), etc.
Degree - The exponent of the highest derivative.
For example, consider the following Differential Equations:
y + $(\frac{dy}{dx})^2$ = 3x
Here, the highest derivative is just first derivative ($\frac{dy}{dx}$). So, order is 1.
The exponent of the highest derivative is 2. So, degree is 2.
$y^2$ + xy + $(\frac{dy}{dx})^4$ + $(\frac{d^3y}{dx^3})^2$ = $e^x$
Here, the highest derivative is third derivative ($\frac{d^3y}{dx^3}$). So, order is 3.
The exponent of the highest derivative is 2. So, degree is 2.
Types of Differential Equations
Ordinary and Partial Differential Equations
One major classification of Differential Equations is based on the number of independent variables in it.
- Ordinary Differential Equations (ODEs) - equations that have only one independent variable (e.g. y).
- Partial Differential Equations (PDEs) - equations that have two or more independent variables.
We will mostly be dealing with Ordinary Differential Equations (ODEs).
Linear and Non-Linear Differential Equations
Differential Equation is linear if no variable or derivative in it has exponent other than one (i.e. no $y^2$, no $\sqrt{y}$, no $\frac{d^2y}{dx^2}$, etc). Also, there should be no functions in it, e.g. cos y, log y, etc.
A linear Differential Equation can only have y, and $\frac{dy}{dx}$.
General form of Linear Differential Equation: $\frac{dy}{dx}$ + F(x) y = G(x)
Here, F(x) and G(x) are functions of x.
For example, 2y + 5$\frac{dy}{dx}$ = 3x
All other differential equations are non-linear differential equations, e.g. Second Order Differential Equations.
Second Order Differential Equations
General form of Second Order Differential Equation: $\frac{d^2y}{dx^2} + F(x) \frac{dy}{dx} + G(x) y = 0$
Second Order Equations maybe of many types, such as homogeneous, non-homogeneous, autonomous, constant coefficients, undetermined coefficients etc.
Bernoulli Equations
General form of Bernoulli Differential Equation: $\frac{dy}{dx} + F(x) y = G(x) y^n$
(Where n can be any real number, except 0 or 1)
When n = 0, then it becomes a Linear Differential Equation.
Homogeneous Equations
$\frac{dy}{dx}$ = F ($\frac{y}{x}$)
In such a case, we can replace $\frac{y}{x}$ by another variable (say v), and then solve it.
For example, $\frac{dy}{dx} = \frac{x^2 + y^2}{xy} = \frac{x}{y} + \frac{y}{x} = (\frac{y}{x})^{-1} + \frac{y}{x}$
$\frac{dy}{dx} = \frac{y}{x} - (\frac{y}{x})^2$