Doubt related to formation of a differential equation

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SUMMARY

The discussion centers on determining the order of the differential equation represented by the function y=C1sin2x+C2cos2x+C3. The consensus is that the order is 2, as the terms are not linearly independent, leading to only two arbitrary constants. The user attempted two methods to derive the differential equation but was confused about the role of C3 as an arbitrary constant. The clarification provided indicates that C3 is not independent due to the linear dependence of the sine and cosine terms.

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cheapstrike
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Homework Statement



Find the order of the differential equation of y=C1sin2x+C2cos2x+C3.

Homework Equations



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The Attempt at a Solution


[/B]
I read in my book that the order of the differential equation is equal to the number of arbitrary constants but the answer given is 2.

Btw I have uploaded two pics with two methods I tried.
1st Pic - http://imgur.com/Py7DTgp
2nd Pic - http://imgur.com/5pdcWq1

In first picture, I calculated upto 3rd differential and obtained a differential equation.

In second picture, I differentiated both sides w.r.t. x and then sent the sin2x term, which I was getting in RHS, to LHS and wrote (1/sin2x) as cosec2x. Then I differentiated both sides again w.r.t. x. In this way, both C1 and C2 which remained after calculating 1st derivative become zero.

Which one is correct method? If it's neither, then what's the right method?
 
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cheapstrike said:
I read in my book that the order of the differential equation is equal to the number of arbitrary constants but the answer given is 2.
Your function only has two arbitrary constants. The terms are not linearly independent.
 
Orodruin said:
Your function only has two arbitrary constants. The terms are not linearly independent.
Can you elaborate please? I mean how is C3 not an arbitrary constant?
 
cheapstrike said:
Can you elaborate please? I mean how is C3 not an arbitrary constant?
The functions that they multiply are linearly dependent. You can rewrite ##\sin^2 x## in terms of cos(2x) and a constant.
 
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Orodruin said:
The functions that they multiply are linearly dependent. You can rewrite ##\sin^2 x## in terms of cos(2x) and a constant.
Thanks
 
cheapstrike said:

Homework Statement



Find the order of the differential equation of y=C1sin2x+C2cos2x+C3.

Can you see how to re-write ##C_1 \sin^2(x) + C_2 \cos 2x + C_3## as ##A \cos2x + B##? Working with the latter form is easier.

Note added in edit: I see that once again PF has fooled me, by only displaying post #4 after I pressed submitted my answer.
 

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