What to observe if unknown coefficients exist?
The differential equation, with two negative coefficients, is unstable so you can expect positive powers in the exponential terms. Check for analysis errors. You can’t simply change the signs of the powers arbitrarily to fit what you believe to be the answer. I suggest you change all signs to positive in the differential equation.

In the op-amp circuit, you should use integrators rather than differentiators, and ensure the in/out signals of the integrators have the correct polarity; if not, invert as appropriate. Also, I can't see where the input cosine signal is applied.