A bit more information is needed to close the loop on this problem. However, some essential parts can be worked out.
First, rearrange the equation to read dy/dt - a y = b
This is called the inhomogeneous equation.
The associated homogeneous equation is dy/dt - ay = 0
This has the solution y = A exp( a t) { presumably a <0 so that it behaves as t ==> ∞ }
Next look for a specific solution to the original equation. That is easily found to be y = -b/a (a constant)
According to the theory of differential equations, the general solution is then
y = A exp(a t) - b/a This can be checked by substitution into the differential equation.
Since we want y to approach 2/3 this means that -b/a = 2/3
Since a is negative, this means that b is positive.
More initial condition information is needed to determine the values for A and a.
