# Differential Equations

## Calculus Worksheets

Home | Online Math Tutoring | online calculus tutoring | **Differential Equations**

### Schedule a Free session to clear worksheet doubts

Have you ever observed what happens when we deposit our money in bank for a certain period of time?

The bank gives us interest on the money we have deposited at a certain rate of interest and we get the total amount which is the sum of our original money deposited i.e. the principal & the interest earned for that period of time.

So, if in a bank principal increases continuously at the rate of 5 % per year, we are interested in finding the worth of the money (say 1000 dollars) after 10 years.

Hence, we require a relation that connects the rate of change of the principal with respect to time i.e. a relation between a dependent variable, independent variable and their rate of change i.e. derivatives.

Here we will introduce the concept of **Differential Equations.**

**I. Definition**

If y=f(x) be a function**, **then** x **is the

**independent**variable &

**is the**

*y***dependent**variable, since its value is dependent on the value/s assigned to

*x*.

In general, an equation which involves **independent variable** (say x),** dependent variable** (say *y*) & the derivatives of the **dependent variable** with respect to **independent variable** is called a **Differential equation**.

Here the derivatives can be of the form ,,,…..

**II. Order of a differential equation.**

**Order** of a differential equation is the **order of the highest order derivative** present in the differential equation. Here the derivative means the derivative of the dependent variable with respect to the independent variable.

means derivative of First order i.e. **order 1.**

means derivative of Second i.e. **order 2.**

means derivative of Third i.e. **order 3.**

**III. Degree of a differential equation.**

**Degree of a differential equation is defined only**, if the given differential equation is a polynomial equation in its derivatives.

**Degree (if defined) **is highest power of the highest order derivative occurring i.e. present in the differential equation.

**Note **

- For a given differential equation
**Order and Degree**(if defined) are both**positive integers i.e. natural numbers**. - For identifying the degree of a differential equation, the derivatives must first be made free from radicals, fractions & then expressed as a polynomial equation in its derivatives. Then we identify its degree.

**EXAMPLES**

Now let’s consider some **examples** on differential equation.

*Example 1*:**Find the order & degree (if defined) of the differential Equation.**

+4 – +2y=0

Here the highest order derivative present in the differential equation is whose order is 3. Hence, its degree is also defined. The highest power of the highest order derivative is one. **Hence the degree is 1.**

*Example 2*: Find the Order & degree (if defined) of the differential Equation.

+ + 7 + =0

Here the highest order derivative present in the differential equation is whose order is 3. Hence the order of the differential equation is 3.

Since, the given differential equation is expressed as a polynomial equation in its derivatives. Hence, its degree is also defined. The highest power of the highest order derivative is four. **Hence the degree is 4.**

*Example 3*: Find the Order & degree (if defined) of the differential Equation.

+ + Cos+5=0

Here the highest order derivative present in the differential equation is whose order is 4. Hence the order of the differential equation is 4.

Since, the given differential equation contains the term Cos, hence it is not a polynomial equation in its derivatives. Hence, **its degree is not defined.**

*Example 4*: Find the Order & degree (if defined) of the differential Equation.

xy – x -y = 0

Here the highest order derivative present in the differential equation is whose order is 2. Hence the **order of the differential equation is 2.**

Since, the given differential equation is expressed as a polynomial equation in its derivatives. Hence, its degree is also defined. The highest power of the highest order derivatives is one. **Hence the degree is 1.**

### Check Point

**Find the order & degree (if defined) of the given differential equations.**

- + 7 – =0
- + – 3 + =0
- x +y=0
- + Sin -12=0
- 2-7+4y=0

#### Answer Key

- Order – 4; Degree – 1
- Order- 4; Degree – 2
- Order – 1; Degree – 1
- Order – 2; Degree – not defined
- Order – 2; Degree – 1

## IN THE NEWS

Home

Fall 2020

Tutoring

Test Prep

Worksheets

Pricing

About Us

Blog

Free Trial

Login

Terms of service

Privacy Policy

Money Back Guarantee

Technical requirements

FAQs

Job Opportunities

Sitemap

**Connect with us**

+1-269-763-4602

+1-269-763-5024

**Online test prep and practice**

SCAT

CogAT

SSAT

ISEE

PSAT

SAT

ACT

AP Exam

**Science Tutoring**

Physics Tutoring

Chemistry Tutoring

Biology Tutoring

**Math Tutoring**

Pre-Algebra Tutoring

Algebra Tutoring

Pre Calculus Tutoring

Calculus Tutoring

Geometry Tutoring

Trigonometry Tutoring

Statistics Tutoring

©2020 eTutorWorld Terms of use Privacy Policy Site by Little Red Bird

©2020 eTutorWorld

Terms of use

Privacy Policy

Site by Little Red Bird