The differential equation is an equation containing one or more derivatives, where derivatives are terms describing the rate of change of quantities that vary continuously. In general, the solution of the differential equation is an equation that expresses the functional dependence of one variable on more than one variable. It typically includes constant terms that are not present in the original differential equation. In applications, functions typically represent physical quantities, derivatives represent their rate of change, and the differential equation determines the relationship between the two. The solution of the differential equation generates a function that can be used to predict the behavior of the original system, at least under some constraints.
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An equation involving a function and its derivatives is a differential equation. Depending on whether partial derivatives are involved or not, it can be referred to as an ordinary differential equation (ODE) or a partial differential equation (PDE). Download the differential equation PDF for free here.
Types of Differential Equations
Here are some of the different types of differential equations:
- Homogeneous Differential Equations
- Nonhomogeneous Differential Equations
- Ordinary differential equations
- Partial differential equations
- Nonlinear differential equations
- Linear Differential Equations
Order of Differential Equation
The order of the highest derivative is the order of a differential equation. Below are some examples for different orders of the differential equation.
(d2y/dx2)+ 2 (dy/dx) + y = 0. The order is 2.
(dy/dt) + y = kt. The order is 1.
First Order Differential Equation
A differential equation of the first order is an equation in which f(x, y) is a function of two variables defined in the XY-plane field. The equation is of the first order since only the first derivative dy/dx is involved (and not higher-order derivatives).
dy/dx = f(x, y) = y’
Second-Order Differential Equation
An equation of the form which is linear in y and its derivatives is called a second-order linear differential equation.
P(x)y”(x) + Q(x)y'(x) + R(x)y(x) = G(x)
Degree of Differential Equation
In the differential equation given, the degree of the differential equation is defined by the power of the highest order derivative. For the degree to be defined, the differential equation must be a polynomial equation in derivatives.
Ex: d4y/dx4 + (d2y/dx2)2 – 3dy/dx + y = 9
Differential Equations Solutions
There are two methods of finding solutions for differential equations:
Separation of Variables: This method is used when the given differential equation can be written in the form of dy/dx = f(y)g(x), where the function f is the function of y only and the function g is the function of x only. Rewrite this problem as 1/f(y)dy = g(x)dx with an initial condition and then integrate it on both sides.
Integrating Factor: This method is used when a given differential equation can be written in the form of dy/dx + p(x)y = q(x) where both the function p and q are functions of x only.
Differential equations of the first order are of the form y’ + P(x)y = Q (x). Where P and Q are the functions of x and the first derivative of y respectively. The differential equation of the higher-order is an equation containing derivatives of an unknown function that can be a partial or ordinary derivative. It can be represented in any order.
- Exponential Decay – Radioactive Material.
- Falling Object.
- Newton’s Law of Cooling.
- RL circuit.
- Exponential Growth – Population.
- Using differential equations, the motion of waves or a pendulum can also be represented.