Help :(

**MissWonder** · September 25th 2009, 08:46 AM

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

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

**Prove It** · September 25th 2009, 08:57 AM

Originally Posted by MissWonder

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

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

The Characteristic Equation is

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

$(m + 2)^2 = 0$

$m = -2$ .

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

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

We can also see that

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

Applying the initial conditions gives

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

Solving the equations simultaneously gives

$2 = 3C_1$

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

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

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

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

**MissWonder** · September 25th 2009, 09:50 AM

Applying the initial conditions gives

and

.

Where do you get that last part from?

**mr fantastic** · September 25th 2009, 05:58 PM

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'.

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