Math Help Please

[werto165]

I'm struggling with what do with a question: 

 

It's 2nd order differential equations. 

 

basically it's the fact that:           y = e^(mx)

differentiate to get:                     y = m*e^(mx)

differentiate again and you get : y= m^2*e^(mx)

 

so here's the question: 

 

Find the general solution to y'' + 4y' + 4y = 0 

 

For the answer I got

 

e^(mx)(m^2 + 4m + 4) =0 

 

so m = -2 

 

so 

 

y = Ae^(-2x) 

 

However the answer says that there is a second term, 

 

ANSWER: 
 

(A + Bx)e^(-2X)

 

How does one get Bx ? I'm really not sure, I thought that I would require more initial condition to work out more terms... 

[Elehat]

1) If you put in any values of A and B it works, so the general solution must contain the (A+Bx)e^.. as a specific solution.

2) The solution space has the correct dimension, so this is the correct answer.

3) The (A + Bx) came about because the solution to the quadratic was a repeated root, so you need a higher dimension term than just the A constant.