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Differential equation is an equation that establishes a relation between functions and the derivatives of these functions. The equation finds a relationship between physical quantities.
- Functions represent the quantities, and their rate of change is represented by the derivatives.
- Differential Equations consist of finding the set of values that satisfy the equation.
- It is used for linking one or more unknown functions and their derivatives.
- The concept is used in the fields of economics, physics and biology.
- Simple differential equations can be solved with formulas.
- An individual can understand the concept by taking the mixture of two solutions.
- Mixing of two salt solutions explains the concept of differential equation.
- Computers are often used to find solutions to complex equations when a solution is unavailable.
Read More: NCERT Solutions For Class 12 Mathematics Chapter 10 Differential Equation
Key Terms: Differential Equations, Functions, Variable, First Order Differential Equation, Derivatives, Second Order Differential Equation, Complex Equations, Polynomial Equation, Dependent variable, Independent variable, Ordinary Differential Equation
What are Differential Equations?
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An equation that contains one or more terms where derivatives of one dependent variable is related with the other independent variable is known as a Differential Equation.
- Partially derivatives and ordinary derivatives are two derivatives present in a differential equation.
- It establishes a relationship between a quantity that is constantly varying with respect to another quantity.
- The value changes, while the rate of change is shown by a derivative.
\(\frac{dy}{dx}\) = f(x)
- In the above equation, “y” is a dependent variable, while “x” is an independent variable.
Read More: Value of Log 1 to 10
Example of Differential EquationExample 1: In Mechanics, it was discovered that the velocity of a freely falling body increases at a rate that is directly proportional to the square root of the vertical distance it covers when it is initially at rest. Since we know the proportionality constant here, it can be expressed mathematically as. v=2gh Now, depending on the hour's need, one can distinguish the expression with respect to time to obtain the relationship between the body's acceleration and velocity, or directly obtain a differential equation in v and h. Example 2: We can take dy/dx = 10x as an example. |
Differential Equations Meaning
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Orders of a Differential Equations
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There are different orders in a differential equation depending on the derivative of that equation. The order of a differential equation is decided by the highest order of the derivative of the equation.
- An order of a differential equation is always a positive integer.
Example of Orders of Differential EquationsExample 1: dy/dx = 5x + 8 , The order is 1. Example 2: y + 2 (dy/dx) + d2y/dx2 = 0. The order is 2. Example 3: kt = dy/dt + y. The order is 1. |
First Order Differential Equation
Derivatives expressed as a linear equation are always in the first order. Only the first derivative as dy/dx is present in such equations, furthermore, x and y are expressed as the two variables:
\(\frac{dy}{dx}\) = f(x, y) = y’
Second-Order Differential Equation
The second-order differential equation is the equation that includes the second-order derivative. It can be expressed as;
\(\frac{d}{dx}\)(\(\frac{dy}{dx}\)) = \(\frac{d^2y}{dx^2}\) = f”(x) = y”
Read More: First Order Differential Equation
Degree of Differential Equations
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When the equation is expressed as a polynomial equation in derivatives like y’,y’’,y’’’, etc, Then the power of the derivative of the highest order is known as the degree of that differential equation. A degree of a differential equations is always a positive integer.
Example of Degree of Differential EquationsExample 1: dy/dx + 2 = 0, Its degree is 1. Example 2: (y”’)3 + 2y” + 6y’ – 16= 0, Its degree is 3. Example 3: (dy/dx) + cos(dy/dx) = 0; Since this equation is not expressed as a polynomial equation in y′, its degree cannot be found. |
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Types of Differential Equations
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There are several types of differential equation which are as follows:
Ordinary Differential Equations
Ordinary Differential Equations is an equation that represents the relation of having one independent variable x, and one dependent variable y, along with some of its other derivatives.
F(\(\frac{dy}{dt}\),y,t) = 0
Partial Differential Equations
A partial differential equation is a type, in which the equation contains many unknown variables with their partial derivatives.
Read More: method of solving a differential equation
Linear Differential Equations
Linear Differential Equations is the linear polynomial equation in which derivatives of several variables are present. It can also be called Linear Partial Differential Equation. There derivatives are partial and function is dependent on the variable.
Homogeneous Differential Equations
When the degree of f(x,y) and g(x, y) is the same, it is known as a homogeneous differential equation.
\(\frac{dy}{dx} = \frac{a_1x + b_1y +c_1}{a_2x + b_2y +c_2}\)
- Due to its simple structure and useful solution, Homogeneous Differential Equation is important in the physical applications of mathematics.
- Since they include homogeneous functions of one form or another, it is critical that we can comprehend homogeneous functions.
Homogenous FunctionsA homogeneous function is one that has multiplicative scaling behavior. It means that when all of its arguments are multiplied by a factor, the function's value is multiplied by the power of that factor.
fαx,αy=anfx,y
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Differential Equation Solution
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The solution can be found by using the following two methods:
Separation of Variables
When the differential equation is written as dy/dx = f(y)g(x). The function f is the function of y only while g is the function of x only. We shall rewrite this as 1/f(y)dy = g(x)dx and then integrate it on both sides after putting an initial condition.
Integrating Factor
When a given differential equation is in the form of dy/dx + p(x)y = q(x) where functions p and q are functions of x only. This is called integrating factor.
Read More: Differentiation and Integration Formula
Applications of Differential Equations
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There are several Differential Equations Applications in the real world, some of them are as follows:
- Various exponential decays and growths can be defined with Differential Equations.
- They are used to describe the change in return on funding over time.
- It is used in the field of medical science.
- It will model cancer growth or the unfolding of ailment within the body.
- Differential Equations help in the movement of power.
- They assist economists in finding the most beneficial investment strategies.
- The motion of waves or a pendulum can also be described using these equations.
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Things to Remember
- Differential Equation establishes a relationship between functions and the derivatives of the functions.
- It establishes a relationship between physical quantities that are represented by functions.
- First-order and second-order are two orders of differential equations.
- It is divided into two categories, namely partial derivatives and ordinary derivatives.
- The equation establishes a relationship between a quantity that constantly varies with respect to another quantity.
- Order is determined by the highest order of the equation.
- It is always a positive integer.
Read More: Applications of Derivatives
Previous Year Questions
- If y(t) is a solution of (1+t)dydt−ty=1 and y(0)=−1, then y(1) is equal to….[JEE Advanced 2003]
- The equation of the curve satisfying the differential equation y(x+y3)dx=x(y3−x)dy and passing through the point (1,1) is….[AMUEEE 2015]
- If xdy=y(dx+ydy),y(1)=1 and y(x)>0. Then, y(−3) is equal to...[JEE Advanced 2005]
- Which of the following is a correct solution of xcosx(dydx)+y(xsinx+cosx)=1?…..[AMUEEE 2014]
- Let f(x) be differentiable on the interval (0,∞) such that f(1)=1 and limt→xt2f(x)−x2f(t)t−x=1 for each x>0 . Then, f(x) is…...[AMUEEE 2014]
- If 8√x(√9+√x)dy=(√4+√9+√x)−1dx,x>0 and $….[JEE Advanced 2017]
- Let y(x) be a solution of the differential equation (1+ex)y′+yex=1. If y(0)=2, then which of the following statements is (are) true ?….[JEE Advanced 2015]
- When y=vx , y and x are variables, the differential equation dydx=2xyx2−y2 reduces to…...[JKCET 2016]
- A particular solution of dydx=(x+9y)2 when x=0,y=127 is..[COMEDK UGET 2007]
- If the firm employees 25 more workers, then the new level of production of items is…..[COMEDK UGET 2013]
- The differential equation which represents the family of curves y=c1ec2xy=c1ec2x, where c1c1 and c2c2 are arbitrary constants is.[AIEEE 2009]
- The curve that passes through the point (2, 3), and has the property that the segment of any tangent to it lying between the coordinate axes is bisected by the point of contact is given by :….[AIEEE 2011]
Sample Questions
Ques. State the usage of differential equations? (2 marks)
Ans. These equations have a top-notch capability to predict the world around us. Furthermore, we use them in a wide type of disciplines, from biology, physics, chemistry, economics, and engineering. Moreover, they are able to describe exponential boom and decay, the populace increase of species.
Ques. Mention the sorts of differential equations? (2 marks)
Ans. The diverse types of differential equations are:
- Ordinary differential equations
- Partial differential equations
- Linear differential equations
- Nonlinear differential equations
- Homogeneous differential equations
- Non-homogeneous differential equations
Ques. State the primary order of the differential equation? (4 marks)
Ans. To begin with, the first-order differential equation is an equation (dy/dx = f(x,y)), in which f(x, y) is a feature of two variables defined on a region in the XY-plane. However, this is a primary-order equation as it entails handiest the first by-product dy/dx (and not higher-order derivatives).
- There are many different types of positions and therefore different activities that can be used to create endless physical processes.
- Differential equations can be categorized in a variety of ways, the most basic of which is based on the order and degree of the difference equation.
- It's a valuable classification because it's easy to find general solutions to differential equations once they've been put in this group.
- It can be generalized, for example, to find the general solution for the second-order differential equation, resulting in the general resolution for the nth order differential equation.
Ques. Why do we want partial differential equations? (3 marks)
Ans. We want to use partial differential equations to mathematically formulate, and consequently help the solution of, physical and different issues that contain functions of numerous variables, which include the propagation of warmth or sound, fluid go with the flow, elasticity, electrodynamics, electrostatics, and so on. Moreover, these mathematical equations call for two or greater impartial variables, an unknown character, and partial derivatives of the regarded function.
Ques. What is the Partial Differential Equation? (4 marks)
Ans. A partial differential equation is a mathematical equation containing, with recognition to independent variables, two or more unbiased variables, an unknown function and partial derivatives of the unknown characteristic.
- The order of the best by-product involved inside the order of the partial differential equation.
- The way to a partial differential equation is a characteristic that solves the equation or, in different phrases, turns it into an identity while substituted into the equation.
- An answer is referred to as popular if it includes all precise answers to the equation concerned.
Ques. Find the differential equation of all the direct lines touching the circle x2 + y2 = r2? (5 marks)
Ans. Let y = mx + c be the equation of all of the direct traces touching the circle.
Given : The equation of the circle is x2 + y2 = r2----------> (1)
The tangent to the circle is c2 = r2(1+m2)
c = r√(1+m2)
we recognise that y = mx + c---------->(2)
y = mx + r√(1+m2) ---------->(three)
y - mx = r√(1+m2)
Differentiating wrt x we get dy/dx -m =0
dy/dx = m
Substituting this in equation (3)
y - (dy/dx . X) = r√(1+(dy/dx)2)
Squaring on each facets, we get
y2 - (dy/dx . X)2 = [ r√(1+(dy/dx)2)]2
[y - x(dy/dx)]2 = r2 (1+(dy/dx))2 is the desired differential equation.
Ques. Determine the general solution to each of the differential equations below:
xdydx=y (log y- log x +1)? (4 marks)
Ans.
Ques. Provided that y=1 when x=1,
x2dy=(2xy+y2)dx
find the specific solution to the differential equation? (4 marks)
Ans.
Ques. The differential equation's degree for 1+\(dy\over dx^2\)=\(d^2y \over dx^2\) is:
a) 4
b) 1
c) 2
d) 3 ? (3 marks)
Ans. Right answer is option b.
Ques. Family y=Ax+A3 of curve represented by the differential equation of degree
a) 3
b) 2
c) 1
d) None of Above? (3 marks)
Ans. Right answer is option a.
Ques. The differential equation's degree for 1+(\(dy\over dx^2\))3/4=(\(d^2y\over dx^2\))1/3 is:
a) 2
b) 4
c) 9
d) 1 ? (3 marks)
Ans. Right answer is option b.
Ques. I.F. of sec x dydx=y+sin x is:
a)sec x
b)e-cox x
c)e-sec x
d)e-sin x ? (3 marks)
Ans. Right answer is option d.
Ques. y satisfies the differential equation:
(4 marks)
Ans. Right answer is option a.
Ques. If sin (x+y)dydx=5, then?
(4 marks)
Ans. Right answer is option a.
Ques. Is it valid that a differential equation is being used? (2 Marks)
Ans. These equations are extremely good at predicting the world around us. They're also used in biology, physics, chemistry, business, and engineering, among other fields. Furthermore, the species' population growth can be described as exponential growth and decay.
Ques. Is the differential equation of the first order? (2 Marks)
Ans. The first-order differential equation is an equation in which f(x,y) refers to two variables specified in the XY plane region. However, since only the first dy/dx derivative is involved, this is a first-order equation (and not higher-order derivatives).
Ques. What is the meaning of partial differential equations? (2 Marks)
Ans. To help solve physical and other problems involving the functions of various variables, such as heat or sound propagation, fluid flow, elasticity, electron dynamics, electrostatics, and so on, by mathematically formulating partial differential equations. Furthermore, these mathematical equations include two or more independent variables, an unknown function, and the partial derivative of the known function.
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