Differential equation example

Differential equation example

DIFFERENTIAL EQUATION

An equation containing an independent variable, a dependent variable and the derivatives of the dependent variable is called a differential equation.

Examples Each of the following equations is differential equation:

(i) dy/dx+ 5y = eˣ

(ii) d²y/dx – dy/dx2 + 3y = sin x

(iii) dy/dx= (x³ – y³)/xy³-x²y

(iv) x²dx + y² dy = 0

ORDER OF A DIFFERENTIAL EQUATION

The order of the highest-order derivative occurring in a differential equation is called the order of the differential equation.

DEGREE OF A DIFFERENTIAL EQUATION
The power of the highest-order derivative occurring in a differential equation, after it is made free from radicals and fractions, is called the degree of the differential equation

Examples
(i) Consider the equation (dy/dx)² + 5y = sin x.
In this equation, the order of the highest-order derivative is 1.
So, its order is 1.
The power of the highest-order derivative is 2.
So, its degree is 2.
Hence, the above equation is of order 1 and degree 2.

(ii) Consider the equation dy/dx² +3(dy/dx)³+2y=0
In this equation, the order of the highest-order derivative is 2.
So, its order is 2.
The power of the highest-order derivative is 1. So, its degree is 1.
Hence, the above equation is of order 2 and degree 1.

(iii) The equation x(d²y/dx²)³ + y(dy/dx)⁴ = 0 is an equation of order and degree 3.

(iv) The equation y dx = x dy may be written as dy/dx=y/x.
So, it is a differential equation of order 1 and degree 1.

SOLUTION OF A DIFFERENTIAL EQUATION
A function of the form y=f(x) +C which satisfies a given differential equation is called its solution.

GENERAL SOLUTION OF A DIFFERENTIAL EQUATION
Suppose a differential equation of order n is being given. If its solution contains n arbitrary constants then it is called a general solution.

PARTICULAR SOLUTION OF A DIFFERENTIAL EQUATION Giving particular values to arbitrary constants in the general solution of a differential equation, we get its particular solutions.

Verify that y= A cos x – B sin x is a solution of the differential equation a

Example

d²y/ dx²+ y = 0 [CBSE 2005C, ’06]

SOLUTION
Given: y=A cosx – B sin x ………. (i)
dy/dx = -A sin x – B cos x
d²y/dx²= -cos x + B sin x
= -(A cos x – B sin x) = -y [from (i)]

d²y/dx²+y=0
Hence,y=Acosx-Bsinx is solution of the differentiol equation d²y/dx²+y=0

Formation of a Differential Equation

whose General Solution is Given

METHOD
Suppose an equation of a family of curves contains n arbitrary constants (called parameters).
Then, we obtain its differential equation, as given below.

Step 1. Differentiate the equation of the given family of curves n times to get n more equations.

Step 2. Eliminate n constants, using these (n + 1) equations. This gives us the required differential equation of order n.

Example
1 – Find the differential equation of the family of curves y = Aeˣ + Be⁻ˣ, where A and B are arbitrary constants.
SOLUTION The equation of the given family of curves is

`` y = Aeˣ + Be⁻ˣ     ......(i)``

Since the given equation contains two arbitrary constants, we differentiate it two times w.r.t. x.
Now,
dy/dx= Aeˣ – Be⁻ˣ

d²y /dx²= Aeˣ+Be⁻ˣ
d²y /dx²-y=0
which is the required differential equation.

2- Find the differential equation of the family of curves y=eˣ(A cos x + B sin x), where A and B are arbitrary constants.

SOLUTION The equation of the given family of curves is y = eˣ(A cos X + B sin x)
Since the given equation contains two arbitrary constants, we differentiate it two times w.r.t. x.
Now,
dy/dx =eˣ(- A sin x + B cos x) + eˣ(A cosx + B sin x)
dy/dx – y=eˣ(-A sinx + B cosx)
d²y/dx²-dy/dx = eˣ (- A cos x – B sin x) + eˣ (- A sin x + B cosx )

d²y/dx²-dy/dx=y(dy/dx-) [using(i)and(ii)]
d²y/dx²-2dy/dx+2y=0

which is the required differential equation of the given family of curves.

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