University:
Georgia Southern University
Course:
MATH 3230 | Ordinary Differential Equations
Academic year:

2024
Views:

395

Pages:

8
Author:

Jalen W.

Property 6: Polynomial Factors Suppose that the Laplace Transform of f (t) exists for s > s0 , and call this F (s). Then, L{t n f (t)}(s) = (−1)n for s > s0 . dn F (s) = (−1)n F (n) (s). ds n Proof We will prove it for n = 1. F 0 (s) = = = = = d [L{f (t)}(s)] ds Z ∞ d −st f (t)e dt ds 0 Z ∞ d f (t)e −st dt ds Z0 ∞ −t · f (t)e −st dt 0 Z ∞ − tf (t)e −st dt 0 = −L{tf (t)}(s) Example Evaluate L{t sin(t)}(s). Example Evaluate L{(t + 1)2 e t }(s). Property 7: Derivatives of Functions Suppose that the Laplace Transform of f (t) exists for s > s0 , and call this F (s). Then, L{f 0 (t)}(s) = sF (s) − f (0) L{f 00 (t)}(s) = s 2 F (s) − f 0 (0) − sf (0) L{f 000 (t)}(s) = s 3 F (s) − f 00 (0) − sf 0 (0) − s 2 f (0) .. . Proof We will show L{f 0 (t)}(s) = sF (s) − f (0). Z ∞ 0 L{f (t)}(s) = e −st f 0 (t) dt |{z} | {z } 0 u Z dv ∞ = uv 0 − v du Z ∞ ∞ −st = e f (t) 0 − f (t) · (−s)e −st dt 0 Z ∞ = [0 − f (0)] + s f (t)e −st dt 0 = −f (0) + sF (s) Example Let Y (s) = L{y (t)}(s). Find the Laplace Transform of the IVP and solve algebraically for Y (s). y 00 + 2y 0 + y = e −t , y (0) = 1, y 0 (0) = 0 Example Let Y (s) = L{y (t)}(s). Find the Laplace Transform of the IVP and solve the corresponding first-order linear ODE for Y (s). ty 00 − ty 0 + y = 2, y (0) = 2, y 0 (0) = −4

Section 8.3: Polynomial Factors and Laplace Transforms of Derivatives

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