1. ## Auxillary Equation

Hi all i need some help with understanding auxillary equation based on boundary values

An example was:

y" + 2y' + y

with y(0) = 1, y(1) = 3

thus putting it into the auxillary equation and finding the roots:

r1 = r2 = -1

putting it into the formula gave

y = C1e^(-x) + C2xe^(-x) note that C1 is C with subscript one and so on :3

Heres the part i dont get. In my book it says

since y(0) = 1 we have 1 = y(0) = C1

Further, as y(1) = 3 we have

3 = y(1) = e^(-1) + C2e^(-1)

So C2 = 3e - 1 and therefore:

y = e^(-x) + (3e-1)xe

Can someone help me with this (could you also explain why C1 disappeared)?

thanks

2. It didn't dissappear. If $\displaystyle C_1=1$, then $\displaystyle C_1e^{-x}=e^{-x}$

-edit- forget the explanations:

$\displaystyle y(0)$ and $\displaystyle y(1)$ are given to know the values of $\displaystyle C_1$ and $\displaystyle C_2$. The solution is fome function $\displaystyle y(x)$ so plugging $\displaystyle x=0$ to obtain $\displaystyle y(0)$ and the same to $\displaystyle x=1$ for $\displaystyle y(1)$, you're able, as you've done, to compute the values of $\displaystyle C_1$ and $\displaystyle C_2$-

3. thanks again mate