general and particular solution

Solving a differential equation does not usually hand you one answer — it hands you a whole family of them, like a stencil that can be slid up and down. The general solution is that entire family, written with arbitrary constants still floating in it. A particular solution is one fixed member of the family, with the constants pinned to numbers. The distinction is the difference between 'the law of all falling bodies' and 'this particular rock, dropped from this ledge at this moment'.

For a first-order equation the general solution carries exactly one arbitrary constant, conventionally written C — for example dy/dx = y has general solution y = C*e^x, a family of exponential curves stacked above and below each other. Fixing C by imposing an initial condition (say y(0) = 3, giving C = 3) selects one particular solution, y = 3*e^x. In general an n-th order equation's general solution carries n arbitrary constants, one for each order, and each side condition you impose consumes one of them. The constants are the recorded 'freedom' left over from integrating.

This is the organizing idea behind almost every solution method you will meet. For linear equations it sharpens into a clean recipe — the general solution is a particular solution plus the general solution of the associated homogeneous equation — which is why the next field opens with 'homogeneous plus particular'. Keeping the bookkeeping straight (how many constants, where they get fixed) is most of what it means to 'solve' a differential equation correctly.