Solutions of a differential equation

A solution of a differential equation is a function y that makes the equation true when you put it in.

To check a claimed solution:

• Differentiate it as many times as the equation needs.
• Substitute y, y′, y″ into the left side.
• Simplify. If you get the right side (often 0), it is a solution.

Example: is y = Ce^(2x) a solution of y′ = 2y? y′ = 2Ce^(2x) = 2y. Yes, for every value of C.

That shows the two kinds of solution:

• General solution: contains arbitrary constants (C, A, B...). The number of constants equals the order of the equation: one for first order, two for second order.
• Particular solution: the constants have been given fixed values, usually from extra information like y(0) = 3. It contains no arbitrary constants.

So y = Ce^(2x) is the general solution of y′ = 2y, and y = 5e^(2x) is a particular solution (the one with y(0) = 5).

Why the count matches the order: each integration brings in one constant, and a second order equation needs two integrations.

If the solution is implicit (x and y mixed, like x² + y² = C), differentiate both sides directly, find y′ from it, and check it fits.