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Hyperreality
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My friend found this problem from Anton
Suppose that the auxiliary equation of the equation
[tex]y'' + py' + qy = 0[/tex]
has a distinct roots [tex]\mu[/tex] and [tex]m[/tex].
(a)Show that the function
[tex]g_\mu(x) = \frac{e^{\mu x} - e^{mx} }{\mu - m}[/tex]
is a solution of the differential equation
(b)Use L'Hopital's rule to show that
[tex]\lim_{\mu\rightarrow\ m} g_\mu(x) = xe^{mx}[/tex]
I tried to proof this using the D-operator method to find the roots, it doesn't seem to work. There seems to be a simpler way of doing this, but I just can't see it.
Any help is appreciated.
Suppose that the auxiliary equation of the equation
[tex]y'' + py' + qy = 0[/tex]
has a distinct roots [tex]\mu[/tex] and [tex]m[/tex].
(a)Show that the function
[tex]g_\mu(x) = \frac{e^{\mu x} - e^{mx} }{\mu - m}[/tex]
is a solution of the differential equation
(b)Use L'Hopital's rule to show that
[tex]\lim_{\mu\rightarrow\ m} g_\mu(x) = xe^{mx}[/tex]
I tried to proof this using the D-operator method to find the roots, it doesn't seem to work. There seems to be a simpler way of doing this, but I just can't see it.
Any help is appreciated.
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