Differential Equation

Differential Equation
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In this section, we will explore the world of differential equations. We will do so by learning about differential equation formulas, methods to solve them, and their applications. You can check out the interactive examples to know more about the lesson and try your hand at solving a few interesting practice questions at the end of the page.

A mathematical equation that includes variables such as x or y, as well as the rate at which those variables change, is a differential equation.

Differential equation

Differential equations are unique since a function rather than a number is itself the solution of a differential equation.

Let's begin and learn more about differential equations.

Lesson Plan

What Is Meant by Differential Equation?

An equation that contains the derivative of a function is called a differential function.

A differential equation is an equation that involves the derivative (derivatives) of the dependent variable with respect to the independent variable (variables) is called a differential equation.

The following are a few differential examples based on trignometric identities and other mathematical concepts.

\(\frac{{dy}}{{dx}} = \sin x\)

\(\frac{{d^2y}}{{dx^2}} + k^2y = 0\)

\(\frac{{d^2y}}{{dt^2}} + \frac{{d^2x}}{{dt^2}} = x\)

\(\frac{{d^3y}}{{dx^3}} + x\frac{{dy}}{{dx}} - 4xy = 0\)

\(\frac{{rdr}}{{d\theta }} + \cos \theta  = 5\)


What Are Different Types of Differential Equations?

Homogenous Differential Equation

A differential equation in which the degree of all the terms is the same is known as a homogenous differential equation.

Example

  1. \(y + x\frac{{dy}}{{dx}} = 0\) is a homogenous differential equation of degree 1.
     
  2. \(x^4 + y^4\frac{{dy}}{{dx}} = 0\) is a homogenous differential equation of degree 4.
     
  3. \(xy\frac{{dy}}{{dx}} + y^2 + 2x = 0\) is not a homogenous differential equation.

Linear Differential Equation

The differential equation of the form \(\frac{{dy}}{{dx}} + Py = Q\) (Where P and Q are functions of x) is called linear differential equation.

\(\frac{{dy}}{{dx}} + Py = Q\) (Where P, Q are constant or functions of y) 

The general solution is \(y\times (I.F.) \) = \(\int {Q(I.F.)dx + c} \) where, I.F(integrating factor) = \(e^\int {^p{^d{^x}}} \) 

Partial Differential Equation

A differential equation that involves partial derivatives is known as a partial differential equation.

Example: \(\frac{{dy}}{{dx}} + x\frac{{dy}}{{dx}} = \frac{{y - t}}{{y + t}}\)

The above differential equation example is a partial differential equation since y  is a function of the two variables x and t, and partial derivatives are present.

Ordinary Differential Equation 

A differential equation that does not involves partial derivatives is known as a partial differential equation.

Example: \(\frac{{d^2y}}{{dx^2}} + \frac{{dy}}{{dx}} = 3y\cos x\)

The above differential equation example is an ordinary differential equation since it does not contain partial derivatives.


How to Find Solution of Differential Equations?

A solution which can be obtained from the general solution by giving particular values to arbitrary constants is called a particular solution.

Therefore the differential equation has infinitely many solutions.

Let us understand differential equation solver by one example

\(\frac{{dy}}{{dx}} = x^2y + y\)

Step 1: Divide above differential equation by y.

\(\frac{1}{y}\frac{{dy}}{{dx}} = x^2 + 1\)

We consider \(y\) and \(x\) both as variables and write this as 

\(\frac{{dy}}{y} = (x^2 + 1)dx\)

Step 2: Now integrate L.H.S. with respect to \(y\) and with respect to \(x\).

\(\int {\frac{1}{y}dx = \int {(x^2 + 1)dx} } \)

Step 3: After integrating, we get:

\(\log y = \frac{{x^3}}{3} + x + c\)

So, this is how differential equation is solved.


What Are Orders of Differential Equation?

The order of a differential equation is the highest order of the derivative appearing in the equation.

Consider the following differential equations,

\(\frac{{dy}}{{dx}} = e^x\), \(\frac{{d^4y}}{{dx^4}} + y = 0\), \((\frac{{d^3y}}{{dx^3}}) + x^2(\frac{{d^2y}}{{dx^2}}) = 0\)

In above differential equation examples, the highest derivative are of first, fourth and third order respectively. 


What Are Degrees of Differential Equation?

The degree of the differential equation is the power of the highest ordered derivative present in the equation.

To find the degree of the differential equation, we need to have a positive integer as the index of each derivative.

Consider the following differential equations examples.

\((\frac{{d^3y}}{{dx^3}})^3 + 2(\frac{{d^2y}}{{dx^2}})^2 - \frac{{dy}}{{dx}} + y = 0\)

In the above differential equation, the highest order term is \((\frac{{d^3y}}{{dx^3}})\) and its power is 3, therefore the degree is 3. 

\((\frac{{dy}}{{dx}})^2 + \frac{{dy}}{{dx}} - \sin^2y = 0\)

In the above differential equation the highest order term is \((\frac{{dy}}{{dx}})\) and its power is 2, therefore the degree is 2.

 
important notes to remember
Important Notes
  1. We can use the following notations for derivatives. \(\frac{{dy}}{{dx}} = y'\),  \(\frac{{d^2y}}{{dx^2}} = y''\),  \(\frac{{d^3y}}{{dx^3}} = y'''\)
  2. The order and degree of a differential equation should be always positive integers.

Solved Examples

Example 1

 

 

What is the order of the differential equation \(\frac{{d^2y}}{{dx^2}} + x(\frac{{dy}}{{dx}}) + y = 2\sin x\).

Solution

The order of the given differential equation  \(\frac{{d^2y}}{{dx^2}} + x(\frac{{dy}}{{dx}}) + y = 2\sin x\) is 2.

\(\therefore \text{The order is 2}\)
Example 2

 

 

What is the degree of the differential equation \(\frac{{d^2y}}{{dx^2}} + \frac{{dy}}{{dx}} + x = \sqrt {1 + \frac{{d^3y}}{{dx^3}}} \)

Solution

The degree of a differential equation is the power of the highest order. 

The degree of the differential equation  \(\frac{{d^2y}}{{dx^2}} + \frac{{dy}}{{dx}} + x = \sqrt {1 + \frac{{d^3y}}{{dx^3}}} \) is one. 

\(\therefore \text{The degree is 1}\) 
Example 3

 

 

Obtain the differential equation for \(y^2 = 4ax\) by eliminating the arbitrary constants.

Solution

\(y^2 = 4ax\).....................(1)

Here, "a" is the arbitrary constant, thus we differentiate with respect to x, 

\(\therefore 2y \frac{{dy}}{{dx}} = 4a\)

Then, statement number (1) becomes: 

\(y^2 = (2y \frac{{dy}}{{dx}})x\) is the required differential equation.

Simplifying it further, we get \(y = (2x \times \frac{{dy}}{{dx}})\).

\(\therefore \frac{{y}}{{2x}} = \frac{{dy}}{{dx}}\). 

Simplifying it further, we get:

\( \frac{{dy}}{{dx}} - \frac{{y}}{{2x}} =0\)

\(\therefore \frac{{dy}}{{dx}} - \frac{{y}}{{2x}} =0 \text{ is the required equation}\)
Example 4

 

 

The rate of decay of the mass of a radio wave substance any time is k times its mass at that time, form the differential equation satisfied by the mass of the substance.

Solution

The rate of decay of mass is \(\frac{{dm}}{{dt}}\).

Here \(\frac{{dm}}{{dt}}\) is directly proportional to m. 

\(\therefore\frac{{dm}}{{dt}} = mk,\text{where k < 0}\)

is the required differential equation.

\(\therefore\ \frac{{dm}}{{dt}} = mk \text{ is the required equation}\)
Example 5

 

 

Reduce the the following differential equation \(1 + \frac{{dy}}{{dx}} = \cos ec(x + y)\).

Solution

\(1 + \frac{{dy}}{{dx}} = \cos ec(x + y)\).............(1)

Put \(x+y=u\)

\(\therefore1 + \frac{{dy}}{{dx}} = \frac{{du}}{{dx}}\)

Given differential equation becomes 

\(\frac{{du}}{{dx}} = \cos ec(u)\)

\(\therefore\frac{{du}}{{\\cosec u}} = dx\)

\(\sin u.du = dx\)

Integrating both sides, we get:

\(\int {\sin u.du = \int {dx} } \)

\(\therefore - \cos u = x + c\)

\(\therefore x + \cos u + c = 0\)

\(\therefore x + \cos (x + y) + c = 0.....(x + y = u)\)

 
Challenge your math skills
Challenging Questions
  1. What is the order and degree of a differential equation \([1 + (\frac{{dy}}{{dx}})^2]^\frac{3}{{2}} = 8\frac{{d^2y}}{{dx^2}}\)?
  2. Find the general solution of a differential equation \(\frac{{dy}}{{dx}} = x\sqrt {25 - x^2} \).

Interactive Questions

Here are a few activities for you to practice.

Select/Type your answer and click the 'Check Answer' button to see the result.
 

 
 
 
 
 

Let's Summarize

We hope you enjoyed learning about the differential equations with the examples and practice questions. Now, you will be able to easily solve problems on differential equation examples, differential equation formulas, and other differential equation applications.

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Frequently Asked Questions (FAQs)

1. What are the real-life applications of differential equations?

There are many real-life applications of a differential equation shown as follows:

  1. In medicine for cancer growth modeling or disease propagation. 
  2. In engineering for the explanation of electricity movements. 
  3. In chemistry for chemical reaction modeling. 
  4. Find optimal investment strategies in economics.  
  5. To explain the motion of waves, pendulums, or chaotic processes in physics.

2. Is differential equation calculus?

Yes, the differential equation is calculus. Differential calculus is a subfield of calculus in mathematics.

3. Why do we use the differential equations?

A differential equation in mathematics is an equation relating to one or more functions and their derivatives. In differential equation applications, the functions typically represent physical quantities, the derivatives represent their rate of change and the differential equation describes the relationship between the two.

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