Differential Equations: Definition, Types, Order and Solved Examples

Differential equation refers to the derivatives of a mathematical equation. In real-world applications, functions represent physical quantities, while derivatives represent the rate at which the function varies with respect to its independent variables. Let's look at the degree and order of differential equations. 

Read More: Differentiation and Integration Formula

Key Terms: Homogeneous differential equations, homogenous functions, Linear differential equations,

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Homogeneous Differential Equations 

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Due to its simple structure and useful solution, the Homogeneous Differential Equation is extremely important in the physical applications of mathematics. The first-order homogeneous differential equation will be discussed in this section. Since they include homogeneous functions of one form or another, it is critical that we first comprehend what homogeneous functions are. So let's get back to work!

The video below explains this:

Differential Equations Detailed Video Explanation:

Also Read:

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Homogeneous Functions 

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A homogeneous function is one that has multiplicative scaling behavior, meaning that when all of its arguments are multiplied by a factor, the function's value is multiplied by the power of that factor. We can assume that a function in two variables, f(x,y), is a homogeneous function of degree n if and only if – 

fαx,αy=anfx,y

where α is a real number. If f(x,y,z) = x² + y² + z² + xy + yz + zx is given, for example. We can note that f(αx,αy,αz) = (αx)²+(αy)²+(αz)²+αx.αy+αy.αz+αz.αx=α²f(x,y,z). As a result, f(x,y,z) is a degree 2 homogeneous function.

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Solved questions examples

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Ques. Determine the general solution to each of the differential equations below:
xdydx=y (log y- log x +1)

Ques. Provided that y=1 when x=1, 
x²dy=(2xy+y²)dx
find the specific solution to the differential equation.

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Linear Differential Equations (LDEs) 

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Linear Differential Equations (LDEs) are a type of differential equation that has 

In the dependent variable y and the independent variable x, a general linear differential equation of order n is an equation with the form – 

a₀(x)dⁿydxⁿ+a¹(x)d^n-1ydx^n-1....................+an-1(x)dydx+an(x)y=b(x)

where a0 is not the same as 0? We will develop a standard method for solving a practical first-order linear differential equation and derive a formula for the general solution in this article. 

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Solved Equations

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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

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

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

Ans.  Right answer is option b.

Because of the diversity of our lives, 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. 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

Ans. Right answer is option d.

Ques. y satisfies the differential equation:

Ans. Right answer is option a.

General and Particular Solutions of a Differential Equation 

Solutions to Differential Equations: A differential equation solution is a relationship between the variables (independent and dependent) that is free of derivatives of any order and satisfies the differential equation in exactly the same way. Let's delve into the specifics of what "differential equations solutions" are.

Ques.  If sin (x+y)dydx=5, then.

Ans. Right answer is option a.

Formation of Differential Equation whose General Solution is Given 

We can detect most physical phenomena, but we can't actually solve the differential equation that's at work. As a consequence, we have the general solution before we even know what equation it is the solution to. Let's start with the basics in order to better understand ordinary differential equations.

Application of Ordinary Differential Equations 

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

Ordinary differential equations 

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. 

Also Read:

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Things to Remember

1. Differential equation refers to the derivatives of a mathematical equation.
2. In real-world applications, functions represent physical quantities, while derivatives represent the rate at which the function varies with respect to its independent variables.
3. Linear Differential Equations (LDEs) are a type of differential equation.
4. A differential equation solution is a relationship between the variables (independent and dependent) that is free of derivatives of any order and satisfies the differential equation in exactly the same way.

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Previous years Questions 

• Then the population of the village at any fixed time tt is given by
• The slope at any point of a curve y=f(x)y=f(x) is given by dydx=3x2dydx=3x2 and it passes through (−1,1)(−1,1) The equation of the curve is​
• The integrating factor of the differential equation dydx+(3x2tan−1y−x3)(1+y2)=0dydx+(3x2tan−1⁡y−x3)(1+y2)=0is​
• The particular solution of the differential equation xdy+2ydx=0, when x=2,y=1 is​
• The integrating factor of the first order differential equation x2(x2−1)dydx+x(x2+1)y=x2−1 is​
• The solution of 25d2ydx2−10dydx+y=0,y(0)=1,y(1)=2e15 is​
• The rate of change of the volume of the cylinder, in cm3/mincm3/min, when the radius is 2cm2cm and the height is 3cm3cm is
• The equation y2+3=2(2x+y)y2+3=2(2x+y) represents a parabola with the vertex at​
• If the coordinates at one end of a diameter of the circle x2+y2−8x−4y+c=0x2+y2−8x−4y+c=0 are (−3,2)(−3,2), then the coordinates at the other end are​
• The general solution of the different equation 100d2ydx2−20dydx+y=0100d2ydx2−20dydx+y=0 is​
• If y′′−3y′+2y=0y″−3y′+2y=0 where y(0)=1y(0)=1, y′(0)=0y′(0)=0, then the value of yy at x=loge2x=loge2 is​
• The degree of the differential equation x=1+(dydx)+12!(dydx)2+13!(dydx)3+.........​
• Solution of the differential equation xdy−ydx=0xdy−ydx=0 represents a
• The equation of one of the curves whose slope at any point is equal to y+2x is​
• The differential equation of all parabolas whose axis is y−axisy−axis is​
• The order and degree of the following differential equation [1+(dydx)2]5/2=d3ydx3[1+(dydx)2]5/2=d3ydx3 are respectively​
• The differential equation of the family of circles passing through the fixed points (a,0)(a,0) and (−a,0)(−a,0) is
• If √y=cos−1x,y=cos−1x, then it satisfies the differential equation (1−x2)−xdydx=c,(1−x2)−xdydx=c, where cc is equal to​