Solve differential equation with variation of parameters

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  • #1
plucker_08
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solve using method of variation of parameters

y''-y = 2/(1+e^x)

y'' ==> second order
 

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  • #2
HallsofIvy
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And your question is?

It should be easy to see that ex and e-x are two independent solutions to y"- y= 0.

The "variation of parameters" is to look for a solution of the form
y(x)= u(x)ex+ v(x)e-x
In fact, there are an infinite number of u(x), v(x) that would work.

Differentiating, y'= u'(x)ex+ u(x)ex+ v'(x)e-x- v(x)e-x.

ASSUME that u'(x)ex+ v'(x)e-x= 0.
(Since there are an infinite number of u(x), v(x) that would work above, this is just "narrowing the search".)

With that assumption y'= u(x)ex- v(x)e-x.

Differentiating again, y"= u'(x)ex+ u(x)ex- v'(x)e-x+ ve-x.

Plug that into the original equation and, because ex and e-x satisfy the original homogeneous equation, the "u(x)" and "v(x)" terms cancel leaving just u'ex- v'e-x= 2/(1+ex). Treat that, along with
u'ex+ v'e-x= 0 (above) as two linear equations for u', v'.
Integrate those solutions to find u and v and plug into y(x)= u(x)ex+ v'(x)e-x.
 

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