Problem in deciding the order of Differential Equation

Question

From this : Doubt related to formation of diff. eq.

Ques:

Let the solution of a differential equation is given as: $y=C_1\sin^2x+C_2\cos 2x+C_3$. Find the order of Differential Equation ??.

This solution satisfies two different ordered differential equation as,

I) $y'''+4y'=0$ (Order 3)

II) $2\cot2xy'+y''=0$ (Order 2)

Out of two answers: Order 2 and Order 3, which solution is correct ?? or both are correct

Please clarify?

Answer 1

There is a trick: the three terms are not linearly independent, only two are, so that the minimum order must be $2$.

Notice that your I) is of order $3$, while your II) is not of order $3$. [Silenty fixed by the OP since.]

Answer 2

You have that $$y=C_1\sin^2x+C_2\cos (2x)+C_3$$ $$y=C_1((1-\cos^2(x))+C_2\cos(2x)+C_3$$ $$y=C_1((1-\frac 12(\cos(2x)+1))+C_2\cos (2x)+C_3$$ $$y=C_5\cos(2x)+C_4$$ It's of order 2 . As Yves Daoust pointed out, equation I) is of order three...