
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) + (3e1)xe
Can someone help me with this (could you also explain why C1 disappeared)? :(
Your contribution is much appreciated.
thanks

It didn't dissappear. If $\displaystyle C_1=1$, then $\displaystyle C_1e^{x}=e^{x}$
(Hi)
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 $\displaystyle C_2$
