On this page, I provide examples of Ordinary Differential Equations, Partial Differential Equations and Linear Differential Equations. I do not normally provide examples, but DFQs require a great deal of time to recognize the various types and the methods to solve them. This is a good starting place for your edification, not for the fainthearted and remember MATH IS FUN.
The applications of partial differential equationshelps in describing various things such as the following:
- In subjects like physics for various forms of motions, or oscillations
- Using differential equations Radioactive decay is calculated.
- The 2nd law of motion by Isaac Newton
- Law of Cooling by Newton
- The wave equation
- Laplace’s equation
- The movement of fluids is described by The Navier–Stokes equations
- For general mechanics, The Hamiltonian equations are used
Differential equations can be divided into several types namely
- Ordinary Differential Equations
- Partial Differential Equations
- Linear Differential Equations
- Nonlinear Differential Equations (List of nonlinear ordinary differential equations)
- Homogeneous Differential Equations
- Nonhomogeneous Differential Equations
Nonlinear Differential Equations
Equations that contain nonlinear terms are known as non-linear differential equations.
All above are nonlinear differential equations. Nonlinear differential equations are difficult to solve, therefore, close study is required to obtain a correct solution. In case of partial differential equations, most of the equations have no general solution. Therefore, each equation has to be treated independently.
Homogeneous Differential Equations
We know that the differential equation of the first order and of the first degree can be expressed in the form Mdx + Ndy = 0, where M and N are both functions of x and y or constants. In particular, if M and N are both homogeneous functions of the same degree in x and y, then the equation is said to be a homogeneous equation. Such an equation can be expressed in the following form:
Thus, a differential equation of the first order and of the first degree is homogeneous when the value of dy/dx is a function of y/x. For example, we consider the differential equation:
Now,
or,
which is equal to a function of y/x
Therefore, the equation
is a homogeneous equation. On the contrary the differential equation
is not a homogeneous equation since in this case, the value of dy/dx is not a function of y/x.
In a homogeneous differential equation, there is no constant term. Whereas, constant terms exist in a linear differential equation. We can find the solution of a linear differential equation if and only if we eliminate the constant term. After removal of the term, we can convert a linear differential equation into homogeneous one. Additionally, variables are absent in homogeneous differential equations having special functions like trigonometric or logarithmic functions.4
Nonhomogeneous Differential Equations
Nonhomogeneous differential equations are the differential equations that contain functions on the right-hand side (R.H.S) of the equations. Nonhomogeneous linear equations have many applications in day-to-day life. Laws of motion, for example, rely on nonhomogeneous differential equations.
We know that homogeneous differential equations are those equations having zero at R.H.S of the equation.
Examples of homogeneous and non-homogeneous differential equations are shown below. We’ll learn how to recognize differential equations based on their form using these examples.5
| Homogeneous Differential Equations | Non-homogeneous Differential Equations |
| y” + y’ – 7y = 0 | y” + y’ – 7y = 8x |
| y” + y’ – y = 0 | y” + y’ – y = 4x2 + 3 |
| y”’ + 5y” – y’ + y = 0 | y”’ + 5y” – y’ + y = 5ex |
Examples of Homogeneous & Non-homogenous Differential Equations
For learning how to solve DFQs, use the Wolfram Alpha Step-by-Step Solutions pages.
- To solve the non-homogenous differential equation, click the link: y”(t) + y(t) = sin(t).
- To solve the homogenous differential equation, click the link: y” + y’ – 7y = 0.
Each page provides a wealth of information, e.g., alternative forms, plot of sample solutions, and a step-by-step instruction on solving the DFQ.
References
4 “Homogeneous Differential Equation: Definition, Methods & Solved Questions”. 2022. Collegedunia. https://collegedunia.com/exams/homogeneous-differential-equation-definition-methods-and-solved-questions-mathematics-articleid-5082.
5 “Nonhomogeneous Differential Equations”. 2022. Codingninjas.Com. https://www.codingninjas.com/codestudio/library/nonhomogeneous-differential-equations.
Example 1 – Ordinary Differential Equation
An ordinary differential equation (also abbreviated as ODE), in Mathematics, is an equation which consists of one or more functions of one independent variable along with their derivatives. A differential equation is an equation that contains a function with one or more derivatives. But in the case ODE, the word ordinary is used for derivative of the functions for the single independent variable.
In case of other types of differential equations, it is possible to have derivatives for functions more than one variable. The types of DEs are partial differential equation, linear and non-linear differential equations, homogeneous and non-homogeneous differential equation.
In mathematics, the term “Ordinary Differential Equations” also known as ODE is an equation that contains only one independent variable and one or more of its derivatives with respect to the variable. In other words, the ODE is represented as the relation having one independent variable x, the real dependent variable y, with some of its derivatives.
y’,y”, ….yn ,…with respect to x.1
Example
Given, y’=2x+1
Now integrate on both sides,
∫ y’dx = ∫ (2x+1)dx
y = 2x2/2 + x + C
y =x2 + x + C
Where C is an arbitrary constant.
References
“Ordinary Differential Equations (Types, Solutions & Examples)”. 2022. BYJUS. https://byjus.com/maths/ordinary-differential-equations/.
Example 2 – Partial Differential Equation
Partial differential equations are equations that consist of a function with multiple unknown variables and their partial derivatives. In other words, partial differential equations help to relate a function containing several variables to their partial derivatives. These equations fall under the category of differential equations.
Partial differential equations are very useful in studying various phenomena that occur in nature such as sound, heat, fluid flow, and waves. In this article, we will take an in-depth look at the meaning of partial differential equations, their types, formulas, and important applications.
Partial differential equations are abbreviated as PDE. These equations are used to represent problems that consist of an unknown function with several variables, both dependent and independent, as well as the partial derivatives of this function with respect to the independent variables.
Example
Numerical methods for partial differential equations
1.1 Finite difference method.
1.2 Method of lines.
1.3 Finite element method.
1.4 Gradient discretization method.
1.5 Finite volume method.
1.6 Spectral method.
1.7 Meshfree methods.
1.8 Domain decomposition methods.
References
commutant. “Partial Differential Equations”. 2022. youtube.com. https://www.youtube.com/playlist?list=PLF6061160B55B0203.
“Partial Differential Equations – Definition, Formula, Examples”. 2022. Cuemath. https://www.cuemath.com/calculus/partial-differential-equations/.
“What Are The Methods For Solving Partial Differential Equations? – MYSQLPREACHER”. 2022. mysqlpreacher.com. https://mysqlpreacher.com/what-are-the-methods-for-solving-partial-differential-equations/.
An introduction to partial differential equations. 17 videos.
Example 3 – Linear Differential Equation
A linear equation or polynomial, with one or more terms, consisting of the derivatives of the dependent variable with respect to one or more independent variables is known as a linear differential equation.
A general first-order differential equation is given by the expression:
dy/dx + Py = Q, where y is a function and dy/dx is a derivative.
The solution of the linear differential equation produces the value of variable y.
Example
Find the solution of the differential equation
Then evaluate
Divide both sides by x2
We have now reduced the above into a linear differential equation of the form
where y is a function, and dy/dx is a derivative. Let’s create and use an integrating factor to make the function integrable.
Integrating tan(u) gives us3
Now, multiply the integrating factor on both the sides of the given differential equation:
Therefore, the general solution of the given differential equation is
calculus4you. “#Q.175 Solution”. 2021. https://www.instagram.com/p/CW8LhkbMB6C/.
“Integration By Parts – Formula, ILATE Rule & Solved Examples”. 2022. BYJUS. https://byjus.com/maths/integration-by-parts/.
Integration by parts is a special technique of integration of two functions when they are multiplied. This method is also termed as partial integration. Another method to integrate a given function is integration by substitution method. These methods are used to make complicated integrations easy. Mathematically, integrating a product of two functions by parts is given as: ∫f(x).g(x)dx=f(x)∫g(x)dx−∫f′(x).(∫g(x)dx)dx
“Linear Differential Equation (Solution & Solved Examples)”. 2022. BYJUS. https://byjus.com/maths/how-to-solve-linear-differential-equation/.
An integrating factor is a function used to solve differential equations. It is a function in which an ordinary differential equation can be multiplied to make the function integrable. It is usually applied to solve ordinary differential equations. Also, we can use this factor within multivariable calculus. When multiplied by an integrating factor, an inaccurate differential is made into an accurate differential (which can be later integrated to give a scalar field). It has a major application in thermodynamics where the temperature becomes the integrating factor that makes entropy an exact differential.2
“Exact Equations Intuition 1 (Proofy) (Video) | Khan Academy”. 2022. Khan Academy. https://www.khanacademy.org/math/differential-equations/first-order-differential-equations/exact-equations/v/exact-equations-intuition-1-proofy.
References
1 “Linear Differential Equation (Solution & Solved Examples)”. 2022. BYJUS. https://byjus.com/maths/how-to-solve-linear-differential-equation/.
2 “Integrating Factor | Solving Differential Equation | Examples”. 2022. BYJUS. https://byjus.com/maths/integrating-factor/.
3 “The Integration Of Tan X – Method, Graph, Definite Integration Of Tan X”. 2022. Cuemath. https://www.cuemath.com/calculus/integration-of-tan-x/.
Additional References
Ryan, Mark. Calculus for Dummies. Indianapolis, Indiana: Wiley Publishing, Inc., 2003.
Differential Equations
“Differential Equations | Khan Academy”. 2022. Khan Academy. https://www.khanacademy.org/math/differential-equations.
“Differential Equations (Definition, Types, Order, Degree, Examples)”. 2022. BYJUS. https://byjus.com/maths/differential-equation/.
“Differential Equations Solution Guide”. 2022. mathsisfun.com. https://www.mathsisfun.com/calculus/differential-equations-solution-guide.html.
“Exact Equations And Integrating Factors”. 2022. mathsisfun.com. https://www.mathsisfun.com/calculus/differential-equations-exact-factors.html.
“How To Solve Differential Equations – wikiHow”. 2022. wikihow.com. https://www.wikihow.com/Solve-Differential-Equations.
A differential equation is an equation that relates a function with one or more of its derivatives. In most applications, the functions represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between them.
In this article, we show the techniques required to solve certain types of ordinary differential equations whose solutions can be written out in terms of elementary functions – polynomials, exponentials, logarithms, and trigonometric functions and their inverses. Many of these equations are encountered in real life, but most others cannot be solved using these techniques, instead requiring that the answer be written in terms of special functions, power series, or be computed numerically.
We say that a differential equation is a linear differential equation if the degree of the function and its derivatives are all 1. Otherwise, the equation is said to be a nonlinear differential equation. Linear differential equations are notable because they have solutions that can be added together in linear combinations to form further solutions.
“Integrating Factor — From Wolfram MathWorld”. 2022. mathworld.wolfram.com. https://mathworld.wolfram.com/IntegratingFactor.html.
“Integrating Factor – Wikipedia”. 2022. en.wikipedia.org. https://en.wikipedia.org/wiki/Integrating_factor.
“Solution Of First Order Linear Differential Equations”. 2022. mathsisfun.com. https://www.mathsisfun.com/calculus/differential-equations-first-order-linear.html.
Partial Differential Equations
“Partial Differential Equations – Definition, Formula, Examples”. 2022. Cuemath. https://www.cuemath.com/calculus/partial-differential-equations/.
Partial differential equations are equations that consist of a function with multiple unknown variables and their partial derivatives. In other words, partial differential equations help to relate a function containing several variables to their partial derivatives. These equations fall under the category of differential equations. Partial differential equations are very useful in studying various phenomena that occur in nature such as sound, heat, fluid flow, and waves. In this article, we will take an in-depth look at the meaning of partial differential equations, their types, formulas, and important applications.
An overview of what ODEs are all about.
The heat equation, as an introductory PDE.
Boundary conditions, and set up for how Fourier series are useful.
Mathematical Mysteries 