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 $D = \dfrac{d}{{dx}}$
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.