The number of arbitrary constants in a particular solution of a differential equation of any order is zero (0) as a particular solution is a solution which contains no arbitrary constant....

The correct option is Option (D) . Since, the number of arbitrary constants ( etc.) in the general solution of a differential equation of  order is 

From eq. (i), =  =  ……….(iii) Putting the values of  and  from eq. (i) and (iii) in L.H.S. of eq. (ii),  =  =  = R.H.S. of eq. (ii) Hence, Function given by eq. (i) is a solution...

Differentiating both sides of eq. (i) w.r.t  we have     = eq. (ii) Hence, Function given by eq. (i) is a solution of 

Differentiating both sides of eq. (i) w.r.t  we have    ……….(iii) Putting the value of  from eq. (i) and value of  from eq. (iii) in L.H.S. of eq. (ii),     = R.H.S. of (ii) Hence, Function given by...

Differentiating both sides of eq. (i) w.r.t  we have        Hence, Function (implicit) given by eq. (i) is a solution of .

 ..(ii)  =  L.H.S. of eq. (ii) =  R.H.S. of eq. (ii) =  =  [From eq. (i)] =  =  =  =  =  L.H.S. = R.H.S Hence,  given by eq. (i) is a solution of .

Differentiating both sides with respect to x, we get, y’ = A Substituting the values of y’ in the given differential equations, we get, \[\begin{array}{*{35}{l}} =\text{ }xy  \\ =\text{ }x\text{...

Differentiating both sides with respect to x, we get, y’ = -sinx Then, Substituting the values of y’ in the given differential equations, we get, \[\begin{array}{*{35}{l}} =\text{ }y\text{ }+\text{...

Differentiating both sides with respect to x, we get, y’ = 2x + 2 Substituting the values of y’ in the given differential equations, we get, \[\begin{array}{*{35}{l}} =\text{ }y\text{ }-\text{...

Differentiating both sides with respect to x, we get, ⇒ y” = ex Then, Substituting the values of y’ and y” in the given differential equations, we get, y” – y’ = ex – ex = RHS. Therefore, the given...