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Unit 9: Solution of Differential Equation
7. The solution obtained from the general solution by giving particular values to the arbitrary Notes
constants, is called a .......................... of the differential equation.
9.3 Solution and Constant of Integration
A solution or integral of a differential equation is a relation between the variables, by means of
which and the derivatives obtained therefore, the equation is satisfied.
Finding the unknown function is called solving or integrating the differential equation. The
solution or integral of a differential equation is also called its primitive, because the differential
equation can be regarded as a relation derived from it.
The solution of a differential equation which contains a number of arbitrary constants equal to
the order of the differential equation is called the general solution (or complete integral or
complete primitive). A solution obtainable from the general solution by giving particular
values to the constants is called a particular solution.
E.g. If y = A cos x + B sin x …………(1)
dy
Then A sin x – B cos x
dx
2
d y
and 2 A cos x – B sin x
dx
2
d y
or 2 y
dx
2
d y
2 + y = 0 …………(2)
dx
Thus (1) is a solution of (2)
Note: Here y = A cos x + B sin x (involving two arbitrary constants A and B) is the general
2
d y
solution of 2 + y = 0 of second order equation.
dx
!
Caution The solution of differential equation of nth order is its particular solution if it
contains less than n arbitrary constants.
Self Assessment
Fill in the blanks:
8. A solution or integral of a differential equation is a .......................... between the variables,
by means of which and the derivatives obtained therefore, the equation is satisfied.
9. Finding the .......................... function is called solving or integrating the differential equation.
10. The solution or integral of a differential equation is also called .........................., because the
differential equation can be regarded as a relation derived from it.
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