Question

The number of arbitrary constant in the particular solution of a differential equation is
$\left(a\right)3$
$\left(b\right)4$
$\left(c\right)$ Infinite
$\left(d\right)$ Zero

Answer 1

Hint – In this question use the basic definition of a particular solution of a differential equation which suggests that all the arbitrary constants must vanish while deriving the particular solution.

Complete step-by-step solution -
Number of arbitrary constants in the general solution of a differential equation is equal to the order of differential equation, while the number of arbitrary constants in a particular solution of a differential equation is always equal to $0$.
Let us consider a differential equation:
$D^{2}y+2Dy+y=e^x$, where $D={\displaystyle \frac{d}{dx}}$
Here the order of the differential equation is 2.
$\therefore$Number of arbitrary constants in the general solution of any differential equation $=$ order of differential equation = 2 , where n is the order of the differential equation.
And the number of arbitrary constants in the particular solution of a differential equation $=0$.
Now we have to find out the number of arbitrary constants in a particular solution of a differential equation.
So according to the above condition it is zero.
Hence, option (D) is correct.

Note – A solution of a differential equation is a function that satisfies the equation. The solution of a homogeneous linear differential equation forms a vector space. In ordinary cases the vector space has finite dimensions equal to the order of equations.