A differential equation is a mathematical equation that contains one or more functions and their derivatives. The function’s derivatives define the rate of change of a function at a given point. It is primarily used in fields such as Physics, Engineering, and Biology. The primary goal of differential equations is to investigate the solutions that satisfy the equations as well as the properties of the solutions. Here’s how to solve differential equations.
Using explicit formulas is one of the simplest ways to solve the differential equation. Let us discuss the definition, types, methods for solving differential equations, order and degree of differential equations, ordinary differential equations with real-world examples, and a solved problem in this article.
Definition of Differential Equation & Derivative Formula
A differential equation is an equation that contains one or more terms as well as the derivatives of one variable (the dependent variable) with respect to another variable (i.e., independent variable)
f(x) = dy/dx
In this case, “x” is an independent variable, while “y” is a dependent variable.
For example, dy/dx = 5x
A differential equation contains derivatives, which can be either partial or ordinary. The derivative represents a rate of change, and the differential equation describes a relationship between one quantity that is constantly changing and another quantity that is changing.
Definition of Derivative Formula
A derivative provides information about the changing relationship between two variables. Take a look at the independent variable ‘x’ and the dependent variable ‘y.’ The derivative formula can be used to calculate the change in the value of the dependent variable in relation to the change in the value of the independent variable expression. The derivative formula can be used to find the slope of a line, the slope of a curve, and the change in one measurement with respect to another measurement.
Formula to Calculate Derivative
A derivative provides information about the changing relationship between two variables. Take a look at the independent variable ‘x’ and the dependent variable ‘y.’ The derivative formula can be used to calculate the change in the value of the dependent variable in relation to the change in the value of the independent variable expression. The derivative formula can be used to find the slope of a line, the slope of a curve, and the change in one measurement with respect to another measurement.
d/dx. (x^n) = n. (x)^(n-1)
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Use of Derivative Formula
In calculus, derivatives are the most important tool. The derivative measures the steepness of a given function’s graph at a specific point on the graph. As a result, the derivative is also known as the slope. It is a ratio of the change in the function’s value to the change in the independent variable. For example, if time is the independent variable, we often think of this ratio as a rate of change, similar to velocity.
Solutions to Differential Equations
A function that solves the given differential equation is referred to as its solution. A general solution is one that contains as many arbitrary constants as the order of the differential equation. A particular solution is one that is free of arbitrary constants. There are two approaches to solving the differential equation.
- Variable separation
- Factor of integration
Applications of Differential Equations
Differential equations are used in a variety of fields, including applied mathematics, science, and engineering. Aside from technical applications, they are also used to solve a variety of real-world problems. Let’s look at some real-time differential equation applications.
- Differential equations are used to describe different exponential growths and decays.
- They are also used to describe how the return on investment changes over time.
- They are used in medical science to stimulate cancer growth or disease spread in the body.
- It can also be used to describe the movement of electricity.
- They assist economists in developing optimal investment strategies.
