Extracting the Fundamental System out of Complex Conjugate Pairs

When dealing with differential equations, particularly arising in the auxiliary equation, it can happen that the homogenous equation- or homogenous counterpart of a nonhoogenous differential equation has solutions that are a complex conjugate such as

$$y\_1=e^{(\alpha +i\beta )x}\text{and}\text{ }y\_2=e^{(\alpha -i\beta )x}$$

Extracting Real Combinations

To extract real valued combinations of these conjugates one forms linear combinations in such a way that the imaginary parts cancel cleanly in one combination (real part) and reinforce in another (imaginary part)

$$y\_R(x)=\frac{y\_1+y\_2}{2}=\frac{e^{ix}-e^{-ix}}{2}=cosx$$ $$y\_I(x)=\frac{y\_1-y\_2}{2i}=\frac{e^{ix-e^{-ix}}}{2i}=sinx$$

General Pattern for any Complex-Conjugate Pair

Whenever one has $y\_1=e^{(\alpha +i\beta )x}$ and $y\_2=e^{(\alpha -i\beta )x}$ The real fundamental system is extractable by the same process. The general formula is

$$y\_R=e^{\alpha x}cos(\beta x)$$ $$y\_I=e^{\alpha x}sin(\beta x)$$