SHAKE TO START!
The first Op Amp is configured as an astable multivibrator, oscillating at 62.5Hz. The voltage at the capacitor is a 4.5V sawtooth wave, which is fed to a buffer. The signal is fed to a step-up transformer, stepping up the voltage to ~120V.
The waveform is a sawtooth one, unlike the usual sine wave from the mains or the square wave of a commercial inverter. This circuit is not intended for commercial applications, but is just a proof of concept.