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William's question at Yahoo! Answers regarding a 3rd order Cauchy-Euler equation

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MarkFL

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Feb 24, 2012
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Here is the question:

There are three linearly independent solutions of the differential equation?


There are three linearly independent solutions of the differential equation t^3y'''+3t^2y''-6ty'+6y=0 of the form t^p. Find the possible values of p.

a. 1,2,3
b. 1,2,-3
c. 1,-2,3
d.1,-2,-3
e. -1,2,3
f. -1,2,-3
g. -1,-2,3
h. -1,-2,-3
I have posted a link there to this thread so the OP can view my work.
 
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MarkFL

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Feb 24, 2012
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Hello William,

We are given the linear ODE:

\(\displaystyle t^3y'''+3t^2y''-6ty'+6y=0\)

We are then told to assume that there are 3 linearly independent solutions of the form:

\(\displaystyle y(t)=t^p\)

Hence:

\(\displaystyle y'(t)=pt^{p-1}\)

\(\displaystyle y''(t)=p(p-1)t^{p-2}\)

\(\displaystyle y'''(t)=p(p-1)(p-2)t^{p-3}\)

Substituting into the ODE, we obtain:

\(\displaystyle t^3p(p-1)(p-2)t^{p-3}+3t^2p(p-1)t^{p-2}-6tpt^{p-1}+6t^p=0\)

Dividing through by \(\displaystyle t^p\ne0\) we obtain the characteristic equation:

\(\displaystyle p(p-1)(p-2)+3p(p-1)-6(p-1)=0\)

\(\displaystyle (p-1)(p(p-2)+3(p-2))=0\)

\(\displaystyle (p-1)(p-2)(p+3)=0\)

Hence, the possible values for $p$ are:

\(\displaystyle p=-3,1,2\)

This is choice b.).