Problem:
y″+4y′+4y=0
Solution:
Note that we do not have the independent variable x, in principle we could have reduce to order to 1, but for this one let's try the solution y=erx. Putting the solution in the problem, we have
r2erx+4rerx+4erx=0
Solving, get r=−2 (repeated).
The repeated solution worried me because then we have only one degree of freedom.
Let's try y=xerx, y′=erx+rxerx, y″=rerx+rerx+r2xerx
y″+4y′+4y=(rerx+rerx+r2xerx)+4(erx+rxerx)+4(xerx)
=erx((r+r+r2x)+4(1+rx)+4(x))
=erx(2r+4+(r2+4r+4))
So if we put r=−2, the equation still work out, so the solution is
Ae−2x+Bxe−2x.
To be honest, I kind of know xe−2x is an answer when I see the double root. I know it because of my experience (sort of my teacher told me so thing). But wouldn't it be more satisfying if we figured out the general rule and why the general rule work? Let's investigate in the next post.
y″+4y′+4y=0
Solution:
Note that we do not have the independent variable x, in principle we could have reduce to order to 1, but for this one let's try the solution y=erx. Putting the solution in the problem, we have
r2erx+4rerx+4erx=0
Solving, get r=−2 (repeated).
The repeated solution worried me because then we have only one degree of freedom.
Let's try y=xerx, y′=erx+rxerx, y″=rerx+rerx+r2xerx
y″+4y′+4y=(rerx+rerx+r2xerx)+4(erx+rxerx)+4(xerx)
=erx((r+r+r2x)+4(1+rx)+4(x))
=erx(2r+4+(r2+4r+4))
So if we put r=−2, the equation still work out, so the solution is
Ae−2x+Bxe−2x.
To be honest, I kind of know xe−2x is an answer when I see the double root. I know it because of my experience (sort of my teacher told me so thing). But wouldn't it be more satisfying if we figured out the general rule and why the general rule work? Let's investigate in the next post.
No comments:
Post a Comment