University:
Stanford University
Course:
MATH 53 | Differential Equations with Linear Algebra, Fourier Methods, and Modern Applications
Academic year:

2023
Views:

26

Pages:

32
Author:

MathPaladin

Questions and Answers Sheet 12 Laplace Transform Question #1 Use Laplace transform method to solve the following initial value problem 𝑑2𝑦 𝑑𝑡 2 + 𝑦 = 𝑒 −𝑡 ; 𝑦(0) = 0, 𝑦 ′ (0) = 0 Answer: Question #2 Use Laplace transform method to solve the following initial value problems 𝑦 ′′ − 3𝑦 ′ + 2𝑦 = 2𝑒 3𝑡 ; 𝑦(0) = 5 and 𝑦 ′ (0) = 7 and show that the answer is 𝑦(𝑡) = 4𝑒 𝑡 + 𝑒 3𝑡 Answer: Question #3 a. Find the Laplace transform 𝐹(𝑠) = 𝐿𝑓(𝑡) of the function 𝑓(𝑡) = 4𝑒 −5𝑡 + 6𝑡 + 4𝑒 5𝑡 , defined on the interval t > 0. 𝐹(𝑠) = 𝐿4𝑒 {5𝑡] + 6𝑡 + 4𝑒 𝑡 b. For what values of s does the Laplace transform exist? Answer: Question #4 Let 𝐺(𝑠) = 𝑡 9 √𝑠 (𝑠+8) be the Laplace transform of the continuous function g(t) and let 𝑓(𝑡) = ∫0 𝑔(𝑥)𝑒 4𝑥 𝑑𝑥 Find the Laplace transform of f(t) 11 a) 𝐹(𝑠) = (𝑠2 +7𝑠)√𝑠−4 9 b) 𝐹(𝑠) = (𝑠+4) √𝑠−4 c) none d) 𝐹(𝑠) = (𝑠2 e) 𝐹(𝑠) = (𝑠2 9 +8𝑠)√𝑠 9 +4𝑠)√𝑠−4 Answer: Question #5 solve for the third order 𝑦 ′′′ − 4𝑦 ′′ − 11𝑦 ′ + 30𝑦 = 4𝑒 8𝑡 + 𝑡 2 using the method of Laplace Transform given that 𝑦(0) = 0, 𝑦 ′ (0) = 0, 𝑦 ′′ (0)? Answer: Question #6 Use Laplace transform method to solve the following initial value problem 𝑑𝑦 𝑑𝑡 4 1 5 5 + 2𝑦 = 4𝑒 3𝑡 subject to the initial condition y(0)=1. show that the answer is 𝑦(𝑡) = ( ) 𝑒 3𝑡 𝑑𝑡 + ( ) 𝑒 −2𝑡 Answer: Question #7 Consider the following initial value problem: 𝑦 ′′ − 6𝑦 ′ − 7𝑦 = sin(2𝑡) 𝑦(0) = 6, 𝑦 ′ (0) = 1 Using Y for the Laplace transform of y(t), i.e., 𝑌 = 𝐿𝑦(𝑡). Find the equation you get by taking the Laplace transform of the differential equation and solve for Y(s) Answer: Question #8 Find 𝐿−1 [ a) 15𝑒 15 6𝑠+7 ] 7𝑡 7 b) 16𝑒 6𝑡 c) 15 6 7 𝑒 −6𝑡 7 d) 6𝑒 6𝑡 Answer: Laplace transformation of 1 𝐿[𝑒 𝑎𝑡 ] = 𝑒 𝑎𝑡 = 𝑠−𝑎 −1 1 𝐿 [ ] 𝑠−𝑎 𝐿−1 [ 15 −1 15 6𝑠+7 𝐿 [ 6𝑠+7 ]= 15 −1 1 𝐿 [ 7] 6 𝑠+ 6 ]= 15 6 𝑒 7 6 − 𝑡 Question #9 1) а) Find the Laplace transform of the function 𝑓(𝑡) = (3𝑡 sin 4 𝑡 + 5𝑒 −4𝑡 + 7𝑡)𝑢𝑠 (𝑡); b) Find the poles and zeros of the transfer function given by 𝑠 2 (𝑠+5) 𝐺(𝑠) = (𝑠+7)(𝑠2 +4𝑠+6) and mark them in the s-plane; c) Find the inverse Laplace transformation of the function given by 𝐺(𝑠) = 2(𝑠+3) 𝑠(𝑠 2 +2𝑠+5) d) Find the initial and final values of the function given in c) Answer: Question #10 Find the inverse Laplace transform 𝑓(𝑡) = 𝐿−1 {𝐹(𝑠)} of the function 𝐹(𝑠) = Answer: 𝑒 −𝑠 (7𝑠−2) (𝑠 2 +25) Question #11 Solve the IVP: 𝑦 ′′ + 3𝑦 ′ + 2𝑦 = 6𝛿6 (𝑡) with conditions: 𝑦(0) = 0, 𝑦 ′ (0) = −1 Answer: Question #12 If f(t) is continuous for 𝑡 ≥ 0, the Laplace Transform of f is the function F defined by ∞ 𝐹(𝑠) = ∫0 𝑓(𝑡)𝑒 −𝑠𝑡 𝑑𝑡 and the domain of F is the set consisting of all numbers s for which the integral converges. Find the Laplace transforms of the following functions. Answer: Question #13 Find the inverse Laplace transform of the following functions: (Note: N is the biggest number from your MEC ID number) 4𝑠 2 −5𝑠+9 a) 𝐹(𝑠) = (𝑠+2)(𝑠2 b) 𝐺(𝑠) = +9) 𝑁𝑠+6 𝑠 2 +9𝑠+5 Answer: Question #14 Find Laplace transform 𝐿{𝑢(𝑡 3 − 1)} Answer: Question #15 solve this equation 𝑦 ′′′ + 2𝑦 ′′ + 4𝑦 ′ + 8𝑦 = 5𝑒 3𝑡 + 1 using Laplace Transformation with initial solution of y(0)=0 , y'(0)=0 , and y''(0)=0. Answer: Question #16 Find the Laplace Transform of the four functions? 𝑓(𝑡) = cos(𝑡) [cos 2(𝑡) − sin2 (𝑡)] cos(3𝑡) Answer: Question #17 Use Laplace transform method to solve the following initial value problem 2𝑦 ′′ + 5𝑦 ′ + 2𝑦 = 𝑒 −2𝑥 ; y(0)=1 and y'(0)=1. Answer: Question #18 Prove using the definition of laplace transform: 𝐿(cos 3 𝑡) = Answer: 𝑠 𝑠 2 +9 Question #19 3𝑠+4 Find the inverse Laplace transform, 𝐿−1 {(𝑠+1)2} Answer: Question #20 Use the table of Laplace transform and properties to obtain the Laplace transform of the following functions. Specify which transform pair or property is used and write in the simplest form. a) 𝑥(𝑡) = cos(3𝑡) b) 𝑦(𝑡) = 𝑡 cos(3𝑡) c) 𝑧(𝑡) = 𝑒 −2𝑡 [𝑡 cos(3𝑡)] d) 𝑥(𝑡) = 3 cos(2𝑡) + 5 sin(8𝑡) e) 𝑦(𝑡) = 𝑡 3 + 3𝑡 2 f) 𝑧(𝑡) = 𝑡 4 𝑒 −2𝑡 Answer: