1550078481-Ordinary_Differential_Equations__Roberts_

Chapter 5 The Laplace Transform Method In sections 4.3 and 4.4 we showed how to solve homogeneous and nonhomo- geneous linear di ...

224 Ordinary Differential Equations The Laplace transform is an operator which assigns to one function, f(x), another function, ...

The Laplace Transform Method 225 EXAMPLE 2 Laplace Transform of x^0 , .C[x^0 ] Calculate the Laplace transform of f(x) = xn, whe ...

226 Ordinary Differential Equations provided s > 0. Now, we use integration by parts a second time by letting u = cos bx and ...

The Laplace Trans! orm Method 227 We now prove the Laplace transform is a linear operator. LINEARITY PROPERTY OF THE LAPLACE TRA ...

228 Ordinary Differential Equations Another property of Laplace transforms which is useful in calculating the transform of some ...

The Laplace Transform Method 229 We now define what it means for a function to be piecewise continuous and calculate the Laplace ...

230 Ordinary Differential Equations If f(x) is piecewise continuous on a finite interval [ a, b] and is continuous except possib ...

The Laplace Transform Method 231 Not all functions have a Laplace transform as the following example shows. EXAMPLE 8 A Function ...

232 Ordinary Differential Equations A function which is of exponential order a > 0 as x ----> +oo may become infinite as x ...

The Laplace Transf arm Method 233 improper integrals. Since Ix: lf(x)le-sx dx converges for s >a, the integral Ix: f(x )csx d ...

234 Ordinary Differential Equations THEOREM 5.2 If f(x) and g(x) are defined and piecewise continuous on [O, b] for all finite b ...

The Laplace Trans! orm Method 235 Table 5 .1 Laplace transforms and inverse Laplace t r ansforms f(x) = .c-^1 (F(s)) F(s) = .C[f ...

236 Ordinary Differential Equations Let f(x) and g(x) be continuous functions on [O, oo) and assume they have Laplace transforms ...

The Laplace Transform Method 237 EXAMPLE 9 Finding an Inverse Laplace Transform 2 For F(s) = s(s + l), calculate .c-^1 [F(s) ]. ...

238 Ordinary Differential Equations Since the linear factor ( s -4) appears to the second power in the denominator of F(s) and t ...

The Laplace Trans! arm Method 239 require using the linearity property or translation property of Laplace trans- forms. To manua ...

240 Ordinary Differential Equations Find .C[h(x)] for l x O-S:x< 2 h(x) = - x ~ 4, 2-:;: x < 4 0, 4-:::: x Find .C[k(x) ...

The Laplace Trans! orm Method 241 The third statement converts the representation of the function g to the proper format for use ...

242 Ordinary Differential Equations 5.2 Using the Laplace Transform and Its Inverse to Solve Initial Value Problems The Laplace ...