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2nd order Linear DE 
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#1
Jan1713, 12:28 PM

P: 203

Hi,
When solving a 2nd order Linear DE with constant coefficients ([itex]ay''+by'+cy=0[/itex]) we are told to look for solutions of the form [itex]y=e^{rt}[/itex] and then the solution (if we have 2 distinct roots of the characteristic) is given by [itex]y(t)=c_1 e^{r_1 t}+c_2 e^{r_2 t}[/itex] This is clearly a solution, but how do we know there are no other solutions? That is, how do we know this is the general solution? 


#2
Jan1713, 02:33 PM

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Thanks
P: 26,148

Hi Apteronotus!
if y'  ry = 0, put y = ze^{rt}, then (z' + rz)e^{rt} = rze^{rt} so e^{rt} = 0 (which is impossible), or z' + rz = rz, ie z' = 0, ie z is constant and now try (y'  ry)(y'  sy) = 0, using the same trick twice 


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