solution of this differential equation which is a very simple version of ”Cruise Control” logic in a car:

Question

x''(t) = −k(x'(t) − y(t)), where x''(t) is acceleration, x'(t) is current speed, y(t) is the desired speed. First, assume that y(t) is constant; then assume that y(t) is a linear function

Answer 1

I give you a hint for the case of $y(t)=c$ being a constant: Differentiate your equation and you obtain $$x'''(t)=-k\cdot x''(t). $$ Now set $x''(t)=z(t)$. Then use seperation by variables (see https://en.wikipedia.org/wiki/Separation_of_variables) to solve the resulting differential equation. Then you integrate $z$ twice to obtain $x$.

If $y$ is a linear function, the same idea can be applied. Just differentiate the differential equation twice.