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Getting repeated root using DSolve for linear 2nd PDE?

Posted 3 months ago
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For a general constant linear 2nd Order pde, like

DSolve[x''[t] + a x'[t] + b x[t] == 0, x[t], t]

the result gives

{{x[t] -> 
   E^(1/2 (-a - Sqrt[a^2 - 4 b]) t) C[1] + 
    E^(1/2 (-a + Sqrt[a^2 - 4 b]) t) C[2]}}

where the repeated root case when b = a^2/4 is ignored, i.e.

DSolve[x''[t] + a x'[t] + b x[t] == 0, x[t], t] /. {b -> a^2/4}

will give

{{x[t] -> E^(-((a t)/2)) (C[1] + t C[2])}}

In general, these two cases are different. So is there a way to use DSolve to give us a conditional expression containing repeated root case?

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