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Variation of Parameters DE

I'm trying to solve the following equation with Variation of Parameters

Y'' + y = 1 = tanx

I understand how to get this far



but I dont understand how they jumped to this step



Can you please show and explain how to go between these two steps? Thanks
Reply 1
Avatar for sarahisme
sarahisme
Hi BENCHMARKMAN,

to jump between the two steps they used the fact that:



where W is the wronskian, and g = 1 + tan(x)

have you come across Green's functions for these types of problems? (it makes things much easier! :P )

try using this:



where W is the wronskian

is this Y'' + y = 1 = tanx

ment to be Y'' + y = 1 + tanx ??
Reply 2
Avatar for BENCHMARKMAN
BENCHMARKMAN
OP
1
Woa you went way over my head. My book or professor has not mentioned wronskian or green functions. I even checked the index of my book to make sure and that word is not in the index. Can you explain what wronskian is? Thanks
BENCHMARKMAN
I'm trying to solve the following equation with Variation of Parameters

Y'' + y = 1 = tanx

I understand how to get this far



but I dont understand how they jumped to this step



Can you please show and explain how to go between these two steps? Thanks

use u=u1 v=u2
u'cosx+v'sinx=0.................(1)
-u'sinx+v'cosx=1+tanx.........(2)

mult (1) by cosx (2) by sin x
u'cos^2x+v'sinxcosx=0
-u'sin^2x+v'sinxcosx=sinx(1+tanx)

subtracting gives
u'=-sinx(1+tanx)
Now
mult (1) by sin (2) by cosx
u'sinxcosx+v'sin^2x=0
-u'cosxsinx+v'cos^2x=cosx(1+tanx)

adding gives
v'=cosx(1+tanx)

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