1. ## Help :(

Okay.. Can anyone explain me, how to solve this problem:

$\displaystyle y''+4y'+4=0$.. when $\displaystyle y(0)=2$ and $\displaystyle y'(0)=0$

2. Originally Posted by MissWonder
Okay.. Can anyone explain me, how to solve this problem:

$\displaystyle y''+4y'+4=0$.. when $\displaystyle y(0)=2 and y'(0)=0$
The Characteristic Equation is

$\displaystyle m^2 + 4m + 4 = 0$

$\displaystyle (m + 2)^2 = 0$

$\displaystyle m = -2$.

Since the solution of the characteristic equation is repeated, the solution will be of the form

$\displaystyle y(x) = C_1e^{-2x} + C_2xe^{-2x}$.

We can also see that

$\displaystyle y'(x) = -2C_1e^{-2x} + C_2e^{-2x} -2C_2xe^{-2x}$.

Applying the initial conditions gives

$\displaystyle 2 = C_1 + C_2$ and $\displaystyle 0 = -2C_1 + C_2$.

Solving the equations simultaneously gives

$\displaystyle 2 = 3C_1$

$\displaystyle \frac{2}{3} = C_1$

$\displaystyle 2 = \frac{2}{3} + C_2$

$\displaystyle \frac{4}{3} = C_2$.

Thus $\displaystyle y(x) = \frac{2}{3}e^{-2x} + \frac{4}{3}xe^{-2x}$.

3. ## hmm

Applying the initial conditions gives

and .

Where do you get that last part from?

4. Originally Posted by MissWonder
Applying the initial conditions gives

and .

Where do you get that last part from?
The reply given was quite clear. Go back and read it carefully. Especially the part where the given initial condition y'(0) = 0 is substituted into the rule for y'.