I Independent functions as solutions

  • Thread starter Sathish678
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Summary
I found three independent functions as solutions for this equation
d/dr(r^2dR/dr) = 6R (cauchy equation)
r^2 , r^(-3) , (1/7)r^6.
I found three independent functions as solutions for this equation
d/dr(r^2dR/dr) = 6R (cauchy equation)
r^2 , r^(-3) , (1/7)r^6.
But i read that a second order linear differential eqn has only two independent solutions.
Why this happened?
 

pasmith

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[itex]\frac17 r^6[/itex] is not a solution. If it were, then as the equation is linear and homogenous [itex]r^6[/itex] must itself be a solution. It is not: [tex]
\frac{d}{dr} \left(r^2 \frac{d}{dr}(r^6) \right) = \frac{d}{dr} \left( 6 r^7 \right) = 42r^6 \neq 6r^6.[/tex]
 

HallsofIvy

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In a bit mote detail, if [itex]R= \frac{1}{7}r^6[/itex] then [itex]\frac{dR}{dr}= \frac{6}{7}r^5[/itex]. So [itex]r^2\frac{dR}{dr}= \frac{6}{7}r^7[/itex] and then [itex]\frac{d}{dr}\left(r^2\frac{dR}{dr}\right)= 6r^6[/itex]. That is NOT equal to 6R because it is missing the "[itex]\frac{1}{7}[/itex]".
 
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