> as lemmas

1834 W. Hamilton in Philos. Trans. (Royal Soc.) 124 250 No general solution has been obtained assigning (as a complete solution ought to do) 3n relations between the n masses.., the 3n varying coordinates.., the varying time t, and the 6n initial data of the problem, namely, the initial coordinates.., and their initial rates of increase..; the quantities called here initial being those which correspond to the arbitrary origin of time.]

1834 W. Hamilton in Philos. Trans. (Royal Soc.) 124 273 The problem of integrating these [differential] equations consists in proposing to assign, by their means, six relations between the time t, the masses m₁m₂, the six varying coordinates..and their initial values and initial rates of increase.

1890 A. R. Forsyth Theory Differential Equations I. iii. 82 He [sc. Jacobi] shewed that the introduction of ‘initial values’ of the variables..renders it possible to take the integrals of the first subsidiary system in a form, which leads immediately to the transformation of the equation.

1902 A. R. Forsyth Theory Differential Equations IV. i. 4 The conditions, as to the arbitrarily assigned values to be acquired at ζ by w and its derivatives, are called the initial conditions; the values are called the initial values.

1920 H. T. H. Piaggio Elem. Treat. Differential Equations iv. 53 As t usually denotes time and x and y rectangular coordinates, a condition such as z = 0 when t = 0 is called an initial condition, while one such as z = 0 if x = 0, or if x = l, or if y = x, is called a boundary condition.

1957 L. Fox Numerical Solution Two-point Boundary Probl. i. 5 Boundary-value problems can always be solved in theory, and often in practice, by a combination of initial-value problems, the extra initial conditions being chosen more or less arbitrarily but finally adjusted to satisfy the prescribed boundary conditions.

1968 L. Fox & D. F. Mayers Computing Methods for Scientists & Engineers iii. 30 We need two conditions, and here there are two main possibilities. In the first, giving the so-called initial-value problem, we are provided with two adjacent values y_s and y_{s + 1} for some s, or possibly y_s and some linear combination of y_{s − 1}, y_s, and y_{s + 1}... The second possibility, giving the so-called boundary-value problem, is the specification say of y₀ and y_n, the values at the two ends of some range.