# The Ordinary Differential Equation, specifying Initial Values for Derivatives of Arbitrary Order, and Boundary Values

## Abstract

The application of the ℒ-transformation in Chapter 15 presupposed the knowledge of the values of the function and its first (*n* − 1) derivatives at *t* = 0. However, one could encounter some initial value problem with *n* specified values at *t* = 0 for derivatives of arbitrary order. For instance, for same third order differential equation one might specify the initial values ^{1} *y*(0), *y* ^{III}(0), *y* ^{IV} (0). In this case, we would solve the problem as if *y* (0), *y*′ (0), and *y*″ (0) were given. Then we would form the higher derivatives, *y* ^{III} (*t*) and *y*^{IV} (*t*). For *t* = 0, we would obtain two linear equations in the unknowns *y*′ (0) and *y*″ (0). Having solved these equations, we can write the complete solution *y*(*t*).

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