Need help! True or false: The only functions with fourth derivative y^(4)=sinx must be of the form y=sinx + c?

This is a true or false problem on a homework assignment of mine. I am inclined to believe it is true, but I don't know how to prove it, and I can't come up with a counterexample to prove it false. Can anyone help?Need help! True or false: The only functions with fourth derivative y^(4)=sinx must be of the form y=sinx + c?
Function


y(x) = sin(x) + x


is also goodNeed help! True or false: The only functions with fourth derivative y^(4)=sinx must be of the form y=sinx + c?
TRUE





The derivative of the sin function is the cos function.





The derivative of the cos function is the sin function.





So any odd numbered derivative of a sin or cos function will be the opposite, for example:





y = sin x





dy/dx = cos x


d3y/dx3 = cos x


d5y/dx5 = cos x


etc.





Conversely any even numbered derivative of a sin or cos function will be the same function, for example:





y = cos x





dy/dx = sin x


d2y/dx2 = cos x


d4y/dx4 = cos x





When working in reverse (i.e. when integrating), you must always add on the unknown constant C that is lost in the differentiation operation.





Edit: ooops, forgot that dy/dx (cos x) = -sin x





Nevertheless, the answer is still true.
Begin with f''''(x) = sin(x) and now integrate four times.





f'''(x) = cos(x) + h





f''(x) = -sin(x) + hx + k





f'(x) = -cos(x) + (h/2)x^2 + kx + u





f(x) = sin(x) + (h/6)x^3 + (k/2)x^2 + ux + v.





Any equation of this form will have a fourth derivative = sin(x). h, k, u, and v are constants, of course.
y^4


4y^3


12y^2


24y





take the derivative four times, this number is like the sinx bit and the +C is the constant which isn't accounted for in the derivative.





i think that's right!


hope it helped!



True





f(x) = sinx


f ' (x) = cos x


f '' (x) = -sin x


f ''' (x) = -cos x


f ';'; (x) = sin x