Problem 1E

In Problems 1–19, use the method of Laplace transforms to solve the given initial value problem. Here x’,y’,etc., denotes differentiation with respect to t; so does the symbol D.

Laplace transforms The formulae of the Fourier transform and the inverse Fourier transform are listed here for reference as 1 ∫ ∞ F(x) = F(!)ei!xd!; 2▯ −∞ ∫ ∞ −i!x F(!) = F(x)e dx: (1) −∞ The Laplace transform can be derived as a special case of the Fourier transforms by limiting the range of f(x) over (0;1). By combining Equation (1) and Equation (1), one obtains ∫∞ (∫∞ ) 1 −i!y i!x F(x) = 2▯ −∞ −∞ F(y)e dy e d!: