Question

If $C_1, C_2, C_3, C_4, C_5$ and $C_6$ are constants, then the order of the differential equation whose general solution is given by $y=C_1 \cos \left(x+C_2\right)$ $+C_3 \sin \left(x+C_4\right)-C_5 e^x+C_6$, is

• A. 6
• B. 5
• C. 4
• D. 3

Answer 1

Answer: (C)

Given,
$y= C_1 \cos \left(x+C_2\right)+C_3 \sin \left(x+C_4\right)+C_5 e^x+C_6$
$y= C_1\left[\cos x \cos C_2-\sin x \sin C_2\right] $
$ +C_3\left[\sin x \cos C_4+\cos x \sin C_4\right]+C_5 e^x+C_6$
$= \cos x\left(C_1 \cos C_2+C_3 \sin C_4\right)$
$ +\sin x\left(-C_1 \sin C_2+C_3 \cos C_4\right)+C_5 e^x+C_6 $
$= A \cos x+B \sin x+C e^x+D$
where, $ A=C_1 \cos C_2+C_3 \sin C_4$
$B=-C_1 \sin C_2+C_3 \cos C_4 \text {, }$
and $ C=C_5, D=C_6$
Hence, order is 4. ( $\because$ number of arbitrary constants is 4)