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