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In this type of problem there is a moment of force at each end acting to keep the beam horizontal i.e the concrete at each end prevents the beam in its immediate neighbourhood from sagging.

If we call this unknown moment of force M then books tend to write this M into the differential equation as POSITIVE and when evaluated it turns out to be NEGATIVE.

Intuitively when you draw a diagram of this case it is clear that M is anticlockwise(downward) and thus negative to prevent the upward curving of the beam ends as it bends downward under it's own weight

So why don't books write this as negative to begin with?

Regards

John