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DSolve with DiracDelta

I tried to solve
DSolve[{y'[t] + \[Alpha] y[t] == DiracDelta[t], y[0] == 0}, y, t]

and got

-\[ExponentialE]^(-t \[Alpha]) (1 - HeavisideTheta[t])

which is zero for t >0.

The solution is correct but I expected somthing else:

exp( - alpha t) UnitStep(t)

which is zero for t <0 and which is also a solution.

In engineering, these solutions are called impulse responses.

The last solution is physically more usefull , because the system responds 
after the excitation (DiracDelta).

How can I force Mathematica to give only solutions with nonzero values for t 
 >0 ?



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