I am solving differential equations via separation of variables. The directions in the problem set say to find the general solution and equilibrium solutions. I realize that some general solutions include the equilibrium solution. However some do not. For example, solving $$\frac{dy}{dt}=2y+1$$ $$\frac{dy}{2y+1}=dt$$ $$\int{\frac{dy}{2y+1}}=\int dt$$ $$\frac{1}{2}\log|2y+1|=t+c$$ $$e^{2t+2c}=2y+1$$ $$y(t)=\frac{e^{2t+2c}-1}{2}$$ Regarding $e^{2c}$ as another constant $k$, $$y(t)=\frac{ke^{2t}-1}{2}$$
Now clearly the equilibrium solution of $-1/2$ does not appear in the expression for the general solution since $ke^{2t}$ would have to equal zero, which does not occur for finite values of $t$. However in the solutions for the problem, it does not include $y(t)=-1/2$ as a solution. What's going on here?
The general solution depends on the constant $k$. The solution $y=-1/2$ corresponds to the particular value $k=0$.