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| If your domain is [0,2] then it is not appropriate to integrate over a delta function with payload at 0. More generally, as you state, an integral containing DiracDelta[x-x0] should not have x0 as an endpoint of the integration range. A way to... |
| I do not know why there is an error but not using [Subscript][1] it works : DSolve[{q[t] + omc*q[t] - omc*q[t - 1] + 2*sig*q'[t] + q''[t] == 0, q[t /; t |