solving initial-value problems by Laplace transform
This is the payoff of the whole subject — the three-step recipe that solves a linear constant-coefficient initial-value problem without ever guessing a form of the solution or matching constants at the end. You transform the problem, solve a piece of algebra, and transform back.
Step one: take the Laplace transform of the entire differential equation, using the derivative rule so that each y', y'' becomes powers of s times Y(s) with the initial values f(0), f'(0) automatically inserted. The differential equation becomes an algebraic equation in Y(s). Step two: solve that algebraic equation for Y(s) — pure algebra, just rearranging. Step three: invert, usually by partial fractions, reading the table backward to recover y(t). Because the initial conditions entered in step one, the answer already satisfies them; there is no separate stage of fixing arbitrary constants.
The method's reach is precisely the linear constant-coefficient case (single equations or systems), and its great strength is uniform handling of forcing terms that defeat the classical methods: discontinuous inputs via the Heaviside step, impulsive inputs via the Dirac delta, and resonant forcing all flow through the same machinery. Its honest limitation is that it does not help with variable-coefficient or nonlinear equations, where the transform of a product is not a product of transforms and the algebra collapses.
For y'' + y = 0, y(0) = 0, y'(0) = 1: transforming gives s^2 Y - 1 + Y = 0, so Y = 1/(s^2 + 1), and inverting gives y(t) = sin(t).
The two initial values are absorbed in the first step, so no constants remain to determine — the inverse transform is the final answer.
The method only works because the equation is linear with constant coefficients; for nonlinear or variable-coefficient equations the transform of the equation is no longer algebraic in Y(s), and the recipe simply does not close.