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)