To prove that the output of a LCCDE is LTI, you need to show that it is both linear and time-invariant.
In order to prove that the output of a Linear Constant Coefficient Differential Equation (LCCDE) is Linear Time-Invariant (LTI), you need to show that the system is both linear and time-invariant.
Linearity can be proven by demonstrating that the equation satisfies the superposition principle. This means that if you have two inputs, x1(t) and x2(t), and their corresponding outputs are y1(t) and y2(t) respectively, then for any constants, a and b, the output of the system with input ax1(t) + bx2(t) should be ay1(t) + by2(t).
Time-invariance can be proven by showing that if the input is delayed by a time shift, the output is also delayed by the same amount of time. In other words, if x(t) produces y(t), then x(t - t0) should produce y(t - t0) for any constant time shift t0.