Why do linear differential equations governing a circuit mean that each circuit element has same angular frequency?
I want to understand why linear differential equations governing a
circuit implies each component has same frequency.

If you differentiate a sine wave you get a cosine wave of exactly the same frequency. No matter how many times you do this, you get the same waveform shape albeit shifted in time and maybe amplitude (but not in frequency). Differentiating does not produce new harmonics when the original signal is sine shaped. To &quot;generate&quot; distortion or non-linearity requires harmonics to be present. It can't happen with linear differential equations.
If you take the series definition of a sinewave for instance: -
$$x - \dfrac{x^3}{3!}+\dfrac{x^5}{5!}-\dfrac{x^7}{7!}+....$$
And then, if you differentiated it you'd get this: -
$$1 - 3\dfrac{x^2}{3!}+5\dfrac{x^4}{5!}-7\dfrac{x^6}{7!}+....$$
Which equals: -
$$1 - \dfrac{x^2}{2!}+\dfrac{x^4}{4!}-\dfrac{x^6}{6!}+....$$
And this is the series definition of a cosine wave. No harmonics are introduced: -