A 5 kHz linear sawtooth generator can be made very simply with a couple of discrete devices:
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The 2N2222 acts as a constant current source to charge the capacitor, which results in a very linear voltage ramp. The SCR discharges the capacitor when the voltage reaches a trigger point set by the pot. I built this circuit to use the absolute minimum number of parts, so the voltage divider pot does double duty to set the transistor bias, and trigger point of the SCR.
Here is the waveform at 5 kHz:
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This circuit would still need an output buffer if you want to feed it into any significant load.
BTW, regardless of which circuit you decide upon, the capacitors must be film type, not ceramic. The capacitance of ceramic capacitors decreases as they charge up, and this noticeably affects linearity of the waveform.