#### Provide solution for RD Sharma maths Class 12 Chapter 21 Differential Equation Exercise Multiple Choice Question Question 9 textbook solution.

Answer : $(c)5$

Hint: The order of differential equation is equal to number of independent arbitrary constants.

Given : $y=C_{1} \cos \left(2 x+C_{2}\right)-\left(C_{3}+C_{4}\right) a^{x+C_{5}}+C_{6} \sin \left(x-C_{7}\right)$

Explanation : $y=C_{1} \cos \left(2 x+C_{2}\right)-C_{3}^{\prime} a^{x} a^{C_{5}}+C_{6} \sin \left(x-C_{7}\right)\; \; \; \; \; \; \; \; \; \; \; \quad\left[C_{3}^{\prime}=C_{3}+C_{4}\right]$

Now $a^{c_{5}}$ is again a constant

$\begin{array}{lr} y=C_{1} \cos \left(2 x+C_{2}\right)-C_{3}^{\prime} a^{x} C_{6}^{\prime}+C_{6} \sin \left(x-C_{7}\right) & {\left[a^{C_{5}}=C_{6}^{\prime}\right]} \\ y=C_{1} \cos \left(2 x+C_{2}\right)-C_{4}^{\prime} a^{x}+C_{6} \sin \left(x-C_{7}\right) & {\left[C_{4}^{\prime}=C_{3}^{\prime} C_{6}^{\prime}\right]} \end{array}$

So the number of independent arbitrary constants are  $C_{1}, C_{4}^{\prime}, C_{2}, C_{6}, C_{7}$

$=5$