notice how the non-homogenous term \(e^x\) on the right hand side of the original DE is a scalar multiple of one of the complementary terms \(Ae^x\),
this corresponds to RESONANCE , so you multiply what you might first think for \(y_p\) by x
\[y_p=(A+Be^x)x\]
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OpenStudy (lgbasallote):
huh?
OpenStudy (anonymous):
The particular solution will be the sum of the particular solutions.
\[y''-y=e^x\] and
\[y''-y=1\]
OpenStudy (lgbasallote):
and that means?
OpenStudy (lgbasallote):
go slow =_=
OpenStudy (lgbasallote):
im not following
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OpenStudy (unklerhaukus):
\[y_p=y_1+y_2\]\[y_{p_1}=A\]\[y_{p_2}=Bxe^x\]
OpenStudy (lgbasallote):
...i dont think that was any slower o.O
OpenStudy (unklerhaukus):
what bit are you stuck on?
OpenStudy (lgbasallote):
just imagine i dont know how to use undetermined coefficients
OpenStudy (lgbasallote):
i dont have much practice yet so im not yet good at it
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OpenStudy (unklerhaukus):
ok so you have a table right/?
OpenStudy (lgbasallote):
table?
OpenStudy (lgbasallote):
maybe showing my solution will help...im not sure if this is right
\[y_p = Ae^x + B\]
\[Dy_p = Ae^x\]
\[D^2 y_p = ae^x\]
\[(D^2 - 1)y = e^x + 1\]
\[Ae^x - (Ae^x + B) = e^x + 1\]
\[-B = e^x + 1\]
\[B = 1 - ^x\]
is this right? if yes what's next?
OpenStudy (lgbasallote):
that should be \[B = 1 - e^x\]
OpenStudy (unklerhaukus):
yeah , that y_p wont work because you already have a e^x term in the complementary solution
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OpenStudy (lgbasallote):
so what's the right way?
OpenStudy (lgbasallote):
\[Ax e^x + B?\]
OpenStudy (unklerhaukus):
yeah i guess that could work, (i havent tried it)
OpenStudy (unklerhaukus):
That should work
OpenStudy (lgbasallote):
so what's next?
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