Solve the initial value problem of the following DE, given that y(0) = 0
dy/dx = (20-2y)/(5-x)
Integrating factor for Linear DE's: e ^ ([tex]\int[/tex]P(x)dx) where Linear DE is of form dy/dx + P(x)y = Q(x).
The Attempt at a Solution
I'm having trouble understanding why my technique doesen't work. I suspect this differential equation needs to be treated as a linear one, but if this is the case I have no idea why it must be treated this way? Why does my approach not work?
Instead of solving for a linear DE, I thought that this equation could be separated:
1/(20-2y) dy = 1/(5-x) dx
integrate both sides:
[tex]\int[/tex]1/(20-2y) dy = [tex]\int[/tex]1/(5-x) dx
-1/2 ln|10-y| = -ln|5-x| + C
ln|10-y| = 2ln|5-x| - 2C
10-y = (5-x)^2 + e^(-2C)
rewrite e^(-2C) as C...
10-y = (5-x)^2 + C
sub in x=0 and y=0
10 = 25 + C
C = -15
10-y = (5-x)^2 - 15
y = 10x - x^2
However my book states that the answer is actually y = 4x - (2/5)x^2
Any help anyone could provide will be very much appreciated. I don't need this done for me, I just want to know why my approach does not work (or what trivial mistake I have made and overlooked), and what alternative approach I should use instead.
Thanks in advance
PS, apologies for the strange formatting, the latex integral signs are oversized, I do not know how to write latex...