# General solution ODE

Consider the following ODE, defined on the interval I = $$\mathbb{R}$$, $$\frac{d^3 y}{dx^3}+\frac{dy}{dx}=x$$ and its homogeneous differential equation, $$\frac{d^3 y}{dx^3}+\frac{dy}{dx}=0$$ whose characteristic polynomial yields the roots $$\lambda_{1} = 0$$, $$\lambda_{2} = i$$ and $$\lambda_{3} = -i$$, each with multiplicity 1. At it this point, there is a fundamental system of complex solutions, $$(\phi_{1}, \phi_{2}, \phi_{3}) = (1, e^{ix}, e^{-ix})$$ and a fundamental system of real solutions, $$(\psi_{1}, \psi_{2}, \psi_{3}) = (1, cos(x), sin(x))$$. Upon applying the undetermined coefficients method, I've come up with a particular solution for the original ODE, $$\eta_{p}(x) = \frac{x^2}{2}$$. Since the coefficients of the given ODE are all real numbers, what is the form of the general solution? Should it be $$\psi (x) = \eta_{p} + c_{1} + c_{2}cos(x) +c_{3}sin(x)$$ or rather $$\phi (x) = \eta_{p} + c_{1} + c_{2}e^{ix} +c_{3}e^{-ix}$$ ? And, in each case, are $$c_{1},c_{2},c_{3} \in \mathbb{R}$$ or $$\mathbb{C}$$? Thank you kindly.

• @RobertThingum They both satisfy the original ODE, so I'm guessing they both work. – minplanck Nov 11 '18 at 0:41

• In this case, the constants $c_{1}, c_{2}, c_{3}$ (which are determined by the initial conditions), should, too, be real, am I right? – minplanck Nov 11 '18 at 0:43