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).
KeywordsImage Function Arbitrary Order High Derivative Order Differential Equation Homogeneous Problem
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