Linearly Dependent Differential Equations

[Weatherkid11]

Show directly that the following functions are linearly dependent on the real line. That is, find a nontrivial linear combination of the funtions that vanishes indetically.
f(x)=17, g(x)= 2sin^2 x, h(x)= 3cos^2 x

Do you just take the 1st and 2nd derivatives and do the determinate?? I am so confused on how to do this problem. Thanks

 

[0rthodontist]

You have to find a nontrivial linear combination of the functions that vanishes identically. In other words, you have to find constants A, B, C, not all zero, so that
A*f(x) + B*g(x) +C*h(x) = 0
Use trig.

 

[Weatherkid11]

So basically something like: 3g(x) + 2h(x) =6sin²x + 6cos²x = 6, but then where do I go from there?

 

[0rthodontist]

Then you use f(x) to make it 0.

 

[Weatherkid11]

ok, I got that A17 has to equal -6 to make it zero, so then that would mean A=-6/17, so the final answer would be (-6/17)17 + 3(2sin²x)+2(3cos²x)=0, Correct? And thanks for the help

 

[0rthodontist]

Yes, correct.