Table of Contents
Significance of Differential Equations
Differential equations are important in mathematics and the sciences because they can be used to model a wide variety of real-world situations. In physics, for example, differential equations can be used to model the motion of particles in a fluid or the trajectory of a projectile. In biology, differential equations can be used to model the growth of populations or the spread of diseases.
The ability to model complex situations using differential equations makes them a valuable tool for scientists and engineers. By solving a differential equation, they can gain a better understanding of how a system behaves and how it might be manipulated to achieve a desired outcome. Additionally, differential equations can be used to predict the future behavior of a system, which can be helpful in designing new technologies or predicting the outcomes of experiments.
Types of Differential Equations Application
There are many different types of differential equations, each with its own unique application. Some of the most common types of differential equations include linear equations, separable equations, and exact equations.
Linear equations are the simplest type of differential equation, and they can be solved using basic algebra. Separable equations can be solved by separating the variables and integrating. Exact equations can be solved using calculus techniques.
Ordinary Differential Equations
An Ordinary Differential Equation (ODE) is a mathematical equation that relates a function and its derivatives. ODEs are used to model the change in a physical quantity over time.
There are many different types of ODEs, but all share a few common features. Each equation typically has a number of variables, which can be either constants or functions of time. The equation also includes a number of derivatives, which are the rates of change of the variables. Finally, each equation includes an initial condition, which is a statement of the value of the variables at some specific point in time.
ODEs can be solved using a variety of methods, including graphical analysis, numerical methods, and the Laplace transform. Once a solution is found, it can be used to predict the behavior of the system over time.
Homogeneous Differential Equation
A differential equation is said to be homogeneous if the derivatives of all the variables in the equation are equal.
For example, the following equation is homogeneous:
$$ y’ = y $$
This equation is true for all values of $ y $.
Nonhomogeneous Differential Equation
A differential equation is said to be nonhomogeneous if it contains a term that is not a function of the independent variable. In other words, a nonhomogeneous equation is an equation that includes a derivative of a variable other than the independent variable.
For example, the equation
is nonhomogeneous because the term is a derivative of .
Partial Differential Equations
A partial differential equation (PDE) is a mathematical equation that relates the rates of change of several variables in different parts of a space or physical system.
PDEs arise in many areas of physics and engineering, including fluid dynamics, elasticity, magnetism, and heat transfer. They can be used to model the behavior of physical systems in a wide range of scientific and engineering applications.
PDEs are commonly expressed in terms of partial derivatives of the variables with respect to space or time. The order of a PDE is the number of variables involved in the equation. A first-order PDE involves one variable, a second-order PDE involves two variables, and so on.
A major challenge in solving PDEs is that they can be extremely difficult to solve analytically. In many cases, numerical methods must be used to approximate the solutions.
Linear Differential Equation
A linear differential equation is an equation in which the highest-order derivative is a linear function of the independent variable. In other words, the equation can be written in the form
\[ y’ = a \left( x \right) + b \left( y \right) \]
where a and b are constants.
Order of Differential Equation
The order of a differential equation is the order of the highest derivative in the equation.
Applications of Differential Equations
Differential equations arise in many scientific disciplines, including physics, chemistry, biology, and engineering. In physics, for example, a differential equation may model the motion of a particle in a given force field. In chemistry, a differential equation may model the rate of a chemical reaction. In biology, a differential equation may model the population dynamics of a species. In engineering, a differential equation may model the vibrations of a structural component.
First Order Differential Equation
A first order differential equation is a mathematical equation that expresses the rate of change of one variable with respect to another. First order differential equations are used to model a wide variety of physical phenomena, including the motion of objects, the change in a population, and the diffusion of molecules.
The most common type of first order differential equation is the linear equation. In a linear equation, the rate of change is proportional to the current value of the variable. This type of equation can be solved using the linear equation solving method.
Nonlinear first order differential equations can also be solved, but this is generally more complicated. In general, nonlinear equations can be solved using a variety of numerical methods, such as the Newton-Raphson Method.
Modeling with First Order Differential Equation
The following example models the population of rabbits over time with a first order differential equation.
\begin{equation}
P(t)=-0.5P(t-1)+4
\end{equation}
The population of rabbits, P(t), decreases by 0.5 each day and increases by 4 each day.
Second-Order Differential Equation
A differential equation of the second order is a differential equation that has two terms in the derivative.
Modeling with Second Order Differential Equation
A second order differential equation is an equation in which the highest power of the independent variable is 2. In other words, the equation involves a derivative of a derivative.
A second order differential equation can be written in the form
\[y”+p(x)y’+q(x)y=f(x)\]
where
y is a function of x
y’ is the derivative of y with respect to x
y” is the derivative of y’ with respect to x
p(x) is a polynomial in x
q(x) is a polynomial in x
f(x) is a function of x
Solved Example
Question: The length of a rectangle is twice the width. If the length is increased by 4 inches, the width will be
1. increased by 2 inches
2. unchanged
3. decreased by 2 inches
4. decreased by 4 inches
