(i) Real functions $y_{2}$ and $y_{2}$ are linearly independent if $Ay_{1}(x)+By_{2}(x)=0$ for all $x$ implies $A=B=0$.

(ii) Suppose that $y_{1}$ and $y_{2}$ are linearly independent solutions of $(*)$. Then the general solution of $(\ast)$ is $y=Ay_{1}+By_{2}$ where $A$ and $B$ are arbitrary real constants. Often this is called the complementary function $(CF)$.

(iii) The auxiliary equation of $(*)$ is the quadratic equation

Given initial conditions $y(0)=C$ and $y^{\prime}(0)=D$, we can choose $A$ and $B$ so that $y=Ay_{1}+By_{2}$ indeed satisfies $y(0)=C$ and $y^{\prime}(0)=D$.