Solution of a differential equation

Checking a solution is a mechanical computation: differentiate the candidate function, substitute it into the equation and verify that both sides agree for every \(x\) in the interval. For instance, \(y=e^{3x}\) solves \(y'=3y\) because \(y'=3e^{3x}\) is exactly \(3y\). One substitution suffices; you need not know how the function was found.

There is rarely a single solution. Since solving a first-order equation essentially amounts to integrating once, an arbitrary constant appears: \(y=x^2+C\) solves \(y'=2x\) for every \(C\). The whole family is called the general solution, and each member, obtained by fixing the constants, a particular solution. For an equation of order \(n\) the family usually carries \(n\) constants.

The word interval matters. A solution is required on an interval where the function and its derivatives exist: \(y=1/(1-x)\) satisfies \(y'=y^2\) but is undefined at \(x=1\), so it is a solution on \((-\infty,1)\) or on \((1,\infty)\), never on all of \(\mathbb{R}\). Even with perfectly regular coefficients, a solution may cease to exist at a finite distance.

Some equations have no real solutions, such as \((y')^2+1=0\), and some have solutions that fit in no family. A common mistake is to believe every differential equation has an explicit formula; in practice many can only be approximated. To pick one specific solution out of the family, extra conditions are needed, which is the topic of the next concept.