# Additional solutions to a given solution to a differential equation

It's my first week studying differential equations, and I would love some feedback regarding the following question:

A second order homogenous linear differential equation with constant coefficients has solution $$cos x$$. Which amongst the following functions have to also be solutions to our function?

$$f_1(x) = \frac{cos x}{3}$$

$$f_2(x) = 1$$

$$f_3(x) = cos x · sin x$$

$$f_4(x) = 3 sin x - 2 cos x$$

$$f_5(x) = cos (x + 2)$$

$$f_6(x) = 0.5x^2 + sin x$$

I have written down: 1, 4 and 5, as they are all versions of the same basic homogenous solution $$y = e^{ax}(c_1cosbx+c_2sinbx)$$. I am just worried I might be missing out on something, as often happens with these kinds of questions.

Many thanks

• $a\ne0$ is not possible.
– user65203
Jun 7, 2019 at 16:15

As a solution is $$\cos x$$, the characteristic polynomial has the roots $$\pm i$$, and the ODE must be
$$y''+y=0.$$
$$C\cos x+S\sin x$$
compatible with $$1,4,5$$.