Question

Find the general solution to the following differential equations
f′′ − 4f′ + 4f = −4 + 12x.

Answer 1

Don't forget to also solve f''-4f'+4f=0 :)

Answer 2

The solution is
\[
f(x) =A e^{2 x}+B x e^{2 x} +3 x+2
\]
To get there first you solve the homegeneous equation
\[
r^2 -4r + 4=(r-2)^2
\]
r = 2 is adouble root.
So the homegeneous equation has the general solution
\[
f(x) =A e^{2 x}+B x e^{2 x}
\]
Now find a particualr solution of the general equation and add it to the other one to obatain
\[
f(x) =A e^{2 x}+B x e^{2 x} +3 x+2
\]

Answer 3

is there a way to do this through linear algebra?

Answer 4

Yes, read
http://tutorial.math.lamar.edu/Classes/DE/SystemsDE.aspx
and try do it yourself.

Answer 5

Let yp= c x + d be a particular solution we need to determine c and d.
yp'= c and yp''= 0 replace in
f′′ − 4f′ + 4f = −4 + 12x. to get
0 - 4c + 4 c x + 4 d = -4 + 12 x
-4 c + 4d + 4 c x = -4 + 4 x
4c =12 ( coef. of x)=> c =3
-4c + 4 d = -4 => 12 + 4 d =-4 => 4 d= 8=> d = 2 so
yp= 3 x + 2