What is the general solution for the given differential equation?

[zak8000]

Homework Statement

y''+4y'+4y= t+exp(-2t)

find the general solution for the differential equation

Homework Equations

The Attempt at a Solution

general solution is sum of complementary function and particular integral

frist finding complementary function

y''+4y'+4y=0

let y=Aexp(mt)

y'=mA=exp(mt)
y''=(m^2)A=exp(mt)

substitute back and get

((m^2)+4m+4)Aexp(mt)=0

m=-2,0

so complementary function:

y=Aexp(-2t)+B

now find particular integral

y''+4y'+4y=t+exp(-2t)

try

y=a+bexp(-2t)
y'=-2bexp(-2t)
y''=4bexp(-2t)

substitute back and get

4bexp(-2t)-8bexp(-2t)+4(a+bexp(-2t))=t+exp(-2t)
(4+4-8)bexp(-2t)+4a=t+exp(-2t) !
so a = t/4 but b will always go to zero i don't know where my mistake is

 

[LCKurtz]

Homework Statement

y''+4y'+4y= t+exp(-2t)

find the general solution for the differential equation

Homework Equations

The Attempt at a Solution

general solution is sum of complementary function and particular integral

frist finding complementary function

y''+4y'+4y=0

let y=Aexp(mt)

y'=mA=exp(mt)
y''=(m^2)A=exp(mt)

substitute back and get

((m^2)+4m+4)Aexp(mt)=0

m=-2,0

Roots are -2, -2.

 

[zak8000]

sorry i do not understand you are you proposing (m^2)+4m+4=0 where m= -2,-2?

 

[LCKurtz]

sorry i do not understand you are you proposing (m^2)+4m+4=0 where m= -2,-2?

What I am saying is you have the roots wrong, hence the solution wrong.

m²+4m+4 = (m+2)²

which has a repeated root of -2. So what is the general solution of the homogeneous equation?