My friend found this problem from Anton(adsbygoogle = window.adsbygoogle || []).push({});

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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# A Tricky Problem

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