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That is, suppose we have a monostable multivibrator and when the DC power is connected, the voltage initially rises rapidly from 0 to V(in). In reality there is a nonzero rise time when you turn on DC power. So the idea is use a schmitt trigger to generate a voltage impulse during the rising time after the power switch is closed and before the maximum DC voltage is reached that will start up the oscillator.

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    \$\begingroup\$ I think you'll attract more answers if you describe why you want to do this (unless it's a purely academic question). Assuming the Schmitt trigger is powered by the same supply, I'm not sure how you're thinking it would be ready to trigger any sooner than the monostable multivibrator. Also note that a monostable multivibrator is not free-running, in the sense of producing sustained oscillation. An astable multivibrator produces a squarish waveform and might be what you're thinking of. \$\endgroup\$
    – scanny
    Mar 16, 2016 at 23:46

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What you are talking about is a POR (Power On Reset) generator, and it's actually quite important. The general, simplest version looks like

schematic

simulate this circuit – Schematic created using CircuitLab This produces a logic LOW upon turnon which depends on the time constant of R1/C1. If you make this time longer than the worst-case risetime of the power supply, you can use it to initialize any circuitry whose starting state you care about. Normally this is applied to logic, but it could just as well be used for analog functions such as astables.

Doing this avoids nasties like unwanted operation of a mechanism or high-voltage output when you turn on the equipment.

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