particular solution
The general solution describes a whole family of curves; a particular solution is one specific member of that family — a single curve, with no leftover adjustable constants. If the general solution is the catalogue, the particular solution is the one item you ordered. It is what you get once the situation has supplied enough information to pin down every constant.
Precisely, a particular solution is obtained from the general solution by assigning definite values to all of its arbitrary constants, usually so as to meet given initial or boundary conditions. Start with y = C e^(kt); if we are told that at time t = 0 the amount is 5, we set t = 0 to get 5 = C e^0 = C, so C = 5, and the particular solution is y = 5 e^(kt). No constants remain free; the curve is now uniquely determined. The data we used to fix the constant is what turns the open-ended family into one concrete answer.
Particular solutions are what real applications actually need: you do not want every possible population curve, you want THE curve that starts from today's population. The general-to-particular move — write the general solution, then apply the conditions to solve for the constants — is the standard final step of nearly every solving method. (Beware a clash of vocabulary later: in the theory of nonhomogeneous linear equations, 'particular solution' takes on a second, narrower technical meaning — any one solution of the forced equation, used as a building block. Same words, related but distinct idea.)
From the general solution y = C e^(kt), the initial condition y(0) = 5 forces C = 5, giving the particular solution y = 5 e^(kt). It is the single curve passing through the point (0, 5) and obeying the equation.
Conditions pin the constants; the family collapses to one definite curve.
Watch for a later double meaning: for nonhomogeneous linear equations, 'particular solution' also names any single solution of the forced equation used to build the general one. Context tells you which sense is meant.