Differential Equations - 15 - Mathematics MCQ

Definition : The equations involving derivatives of one or more dependent variables with respect to a single independent variable are called ordinary differential equations.
Ex.:

are the examples of ordinary differential equations where y is dependent variable and x is independent variable.

Definition : The order of the highest ordered derivative involved in a differential equation is called the order of the differential equation. Thus in Q. 1 :
The order of DE(1) = 3
The order of DE(2) = 2
The order of DE(3) = 1

Note that
Order of DE in (a) = 2
Order of DE in (b) = 1
Order of DE in (c) = 3

Note that
Order of DE in (a) = 1
Order of DE in (b) = 2
Order of DE in (c) = 2

Definition : The degree of a differential equation is defined as the highest power of the highest order derivative involved in the differential, where the equation has been made rational and integral as far as the derivatives are concerned. 
degree of DE (1) = 2
degree of DE (2) = 1
degree of DE (3) = 1
An Important Example : Find the degree of the differential equation.

In order to rationalize this equation, we first square this equation and get

Now clearly the degree of the differential equation is 2.
Thus statement (b) gives the correct definition of the degee of a differential equation.

Degree of this differential equation ean not be defined.

It is clear from the definition that the order of a differential euation is unique.

i. Clearly degree of DE in (a) = 1
ii. DE in (b) must be first squared to make it. rational. Then differential equation becomes.

or

so that the highest power of the highest order derivative is 2. Hence its degree is 2.
iii. The differential equations in (c) and (d) must be squared first to make them rational.
Then we find that their degree is 1.
∴ The differential equations in (a), (c) and (d) are each of degree 1 and in (b) is of degree 2.

dy/dx + 1 = cosec (x + y)
Let x + y = t and 1 + dy/dx = dt/dx
⇒ dt/(cosec t) = dx
∴ ∫ sin t dt = ∫ dx
⇒ − cos t = x − c
⇒ cos(x + y) + x = c

See the definition of the degree of a differential equation.

The degree of an equation that has not more than one variable in each term is the exponent of the highest power to which that variable is raised in the equation. But when more than one variable appears in a term, it is necessary to add the exponents of the variables within a term to get the degree of the equation. Hence, the degree of the equation, x³ - 6x³ y³ + 2xy = 0, is 3 + 3 = 6.

Rewrite the given differential equation as

Then clearly, its order is 3 and degree is 2.

Here m = n = 2

DE (a) is of order 2 and degree 1
DE (b) is of order 1 and degree 2
DE (c) is of order 1 and degree 1
DE (d) is of order 2 and decree 1

Verify by actual substitution of y = sinx and y = cos x in the given differential equation.

Some Definitions.
General solution : A solution of the differential equation of order n containing n essential arbitrary constants is called a general solution of the differential equation.
Particular solution : A solution which is obtained from the general solution by giving particular values to the arbitrary constants is called a Particular solution.
Singular solution : A solution which can not be obtained from the general solution of the differential equation by any chioice of the n essential arbitrary constants is ealled a Singular solution.
Ex : Consider the differential eauation.
 ..(i)
We have
General solution :    y = (x + c)²
Particular solutions : y = x : For c = 0
y = x + 10 : For c = 10 etc.
Singular solutions : y = 0 Note that y = 0 satisfies the differential equation (i) and can not be obtained from the general solution by giving any value to c.
The given differential equation is


Its general solution is given by

But the solution y² - 20x (verify that it satisfies equation (ii)) but it can not be obtained from the general solution given by (iii) for any value of c. Therefore, y² -= 20x is a singular solution of differential equation (ii).

We shall explain the method’ of obtaining the differential equation satisfied by a given family of curves.
The method is to eliminate the constants in the family (these constants arc known as the parameters of the family).
Solution : Ex. 1 : Find the differential equation satisfied by
y = cx + c    ...(i)
for different values of c.
Solution : Differentiating (i) we get
dy/dx = c   .....(ii)
Eliminating e from (i) and (ii)

Example : 2 : Determine the differential equation whose general solution is
y = Ae^2x + Be^2x ...(iii)
Solution : Defferendating equation (iii), we get

Differentiating (iv) again,

∴ Required differential equation is

Remark : Note that you have to differentiate the given equation of the family of curves as many times as the number of arbitrary constants occuring in the given equation.