Find an initial value problem whose unique solution is the...

Find an initial value problem whose unique solution is the given function, and then use it to compute the Laplace transform of the function (see Example 10). Take $a$ to be a nonzero real number.
(a) sinh af.
(b) coshat.

Key Concepts

Laplace Transform

The Laplace transform is an integral transform that converts functions of time into functions of a complex variable. It is widely used to solve differential equations by turning them into algebraic equations. In this context, the Laplace transform is applied to functions that solve an initial value problem, and its properties—such as linearity and the transform of derivatives—make it a powerful tool for solving and analyzing the behavior of systems described by differential equations.

Initial Value Problems

An initial value problem (IVP) consists of a differential equation along with specified values of the unknown function and, possibly, its derivatives at a given point, typically at time zero. The IVP framework ensures the existence and uniqueness of the solution under appropriate conditions. Here, the given function is recognized as the unique solution to an IVP, which is used to derive its Laplace transform.

Hyperbolic Functions

Hyperbolic functions, such as sinh and cosh, are analogues of the trigonometric functions but for a hyperbola, defined in terms of exponential functions. They have distinctive derivative properties—namely, the derivative of sinh is cosh, and the derivative of cosh is sinh—which are essential for forming differential equations that these functions satisfy. These properties allow one to derive the differential equation whose solution is the given hyperbolic function, facilitating the application of the Laplace transform.

Derivation of Differential Equations from Known Solutions

Given a function known to be the solution of an IVP, one can recover the differential equation it satisfies by computing its derivatives and eliminating arbitrary constants. This derivation is useful because it links the behavior of the function to its governing equations and provides a structured approach to compute transforms by leveraging the relationship between derivatives in the time domain and algebraic manipulation in the Laplace domain.

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