Page 119 - DMTH202_BASIC_MATHEMATICS_II

Basic Mathematics-II

Notes Substituting these values in the given equation,
2
d y
 y  (A cosx + B sinx) + A cosx + B sinx = 0.
dx 2

Notes

1. The solution of a differential equation is also called a primitive.
2. The order of the differential equation determines the number of arbitrary constants
in the solution.

9.2.1 General Solution

The solution of a differential equation, in which the number of arbitrary constants is equal to the
order of the differential equation is called the general solution or complete solution or the
complete primitive.
x
x
For example, y = C e + C  e involving two arbitrary constants C and C is the general solution
1 2 1 2
2
d y
of the differential equation  y  0 of second order.
dx 2
2
d y
Similarly, the solution y = A cosx + B sin x obtained for the differential equation 2  y  0 is
dx
the general solution.
9.2.2 Particular Solution

The solution obtained from the general solution by giving particular values to the arbitrary
constants, is called a particular solution of the differential equation.

For example, y = cos x (taking A = 1 and B = 0) and y = sin x (by taking A = 0 and B = 1) are some
2
d y
particular solutions of the differential equation  y  0 .
dx 2

!
Caution Different techniques can be used to find the solutions of differential equations of
first order and first degree depending on type of given differential equation.

Self Assessment

Fill in the blanks:
5. Any relation between the dependent and independent variables, when substituted in the
differential equation, reduces it to an identity is called a solution or .......................... of the
differential equation.
6. The solution of a differential equation, in which the number of arbitrary constants is equal
to the order of the differential equation is called the .......................... .

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