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Order of DE for y=c₁cos2x+c₂cos²x+c₃sin²x+c₄ | JEE

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Question
The order of the differential equation corresponding to y=c1cos⁡2x+c2cos⁡2x+c3sin⁡2x+c4y = c_1\cos 2x + c_2\cos^2 x + c_3\sin^2 x + c_4 is:
A22correct
B44
C33
D11
Solution
Use cos⁡2x=1+cos⁡2x2\cos^2 x = \dfrac{1+\cos 2x}{2} and sin⁡2x=1−cos⁡2x2\sin^2 x = \dfrac{1-\cos 2x}{2}:
y=c1cos⁡2x+c22(1+cos⁡2x)+c32(1−cos⁡2x)+c4.y = c_1\cos 2x + \frac{c_2}{2}(1+\cos 2x) + \frac{c_3}{2}(1-\cos 2x) + c_4.
 Collect the constant and the cos⁡2x\cos 2x terms:
y=(c22+c32+c4)+(c1+c22−c32)cos⁡2x=A+Bcos⁡2x.y = \left(\frac{c_2}{2}+\frac{c_3}{2}+c_4\right) + \left(c_1 + \frac{c_2}{2} - \frac{c_3}{2}\right)\cos 2x = A + B\cos 2x.
 ⇒\Rightarrow only two independent arbitrary constants A,BA, B survive, so the order is 22. Correct answer: (1)
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