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1. Use the linearity of property of Laplace transforms and the table of Laplace transforms on MyUni to find the Laplace transform of v(t) = av (t) + Bv2(t), where vi(t) = 6(t - 2) and v2(t) = u(t – 3). Here, a and 8 are constants, and ő(t) and u(t) are the Dirac delta function and unit step function.

1. Use the linearity of property of Laplace transforms and the table of Laplace transforms on MyUni to find the Laplace transform of v(t) = av (t) + Bv2(t), where vi(t) = 6(t - 2) and v2(t) = u(t – 3). Here, a and 8 are constants, and ő(t) and u(t) are the Dirac delta function and unit step function.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Uh need to solve only (question 1) I also attached the table that is given in questions to solve uh only need to solve the q1 in one hour and get thumb up plz plz plz solve perfectly and handwritien soloution needed plz
| 25 |** | OLt - a) 1:4 1:4 Table of Laplace Transforms F(s) = L{S(t)} f(t) 1/s 1/s² 1/s" (n = 1,2, ...) | 1/V 1/s3/2 1/s (a > 0) 1 1 t"-1/(n – 1)! 1//mt 3 4 Basic General Formulas for the Laplace Transform 6. t0-1/T(a) 1 Formula Name, Comments eat 8- a 1 F(s) = L{S(1} = [e"()dt Definition of Transform teat (8 – a)² 1 | (1) = c-'{F(»)} Inverse Transform 9. (8 – a)™ 1 (n = 1, 2, .) (n-1)! 1 (eat – e") C{af(t) + bg(t)} = aL{f(t)} + bL{g(t)} Linearity 1 (a # b) 10 (8 – a)(s – b) (a – b) 1 (aet – be") - sCf) – f(0) (a ± b) 11 (8 – a (s – b) (a – b) Differentiation of Function - PL(S) – sf(0) – s'(0) 12 sin wt s² + w? cism) = "L(J) – *-'s(0) – .. – fln-1)(0) 13 Cos wt C{e“{(t)} = F(s – a) -Shifting a 14 sinh at L-{F(s – a)} = e" S() (Ist Shifting Theorem) 15 cosh at L{{(t – a)u(t – a)} = e~**F(s) t-Shifting at sin wt 16 (8 – a)² + w² {e"F(s)} = f(t – a)u(t – a) (2nd Shifting Theorem) 8-a 17 eat cos wt (S • 9)(t) = [(r)a(t – r)dr (6 – a)² + w² Convolution 1. (1 – cos wt) 18 s(s² + w²) CJ • 9) - C)C() 19 (wt – sin wt) 1 (sin wt – wt cos wt) 2 1 20 (8² + w?)Z sin wt 21 (s² + w?)Z 1 (sin wt + wt coswt) 22 (s² + wa )Z (a² # b³) 1 (cos at - cos bt) 23 (s² + a²)(s² + b²) 62 - a 24 u(t – a) 25 eas 6(t – a)
Transcribed Image Text:| 25 |** | OLt - a) 1:4 1:4 Table of Laplace Transforms F(s) = L{S(t)} f(t) 1/s 1/s² 1/s" (n = 1,2, ...) | 1/V 1/s3/2 1/s (a > 0) 1 1 t"-1/(n – 1)! 1//mt 3 4 Basic General Formulas for the Laplace Transform 6. t0-1/T(a) 1 Formula Name, Comments eat 8- a 1 F(s) = L{S(1} = [e"()dt Definition of Transform teat (8 – a)² 1 | (1) = c-'{F(»)} Inverse Transform 9. (8 – a)™ 1 (n = 1, 2, .) (n-1)! 1 (eat – e") C{af(t) + bg(t)} = aL{f(t)} + bL{g(t)} Linearity 1 (a # b) 10 (8 – a)(s – b) (a – b) 1 (aet – be") - sCf) – f(0) (a ± b) 11 (8 – a (s – b) (a – b) Differentiation of Function - PL(S) – sf(0) – s'(0) 12 sin wt s² + w? cism) = "L(J) – *-'s(0) – .. – fln-1)(0) 13 Cos wt C{e“{(t)} = F(s – a) -Shifting a 14 sinh at L-{F(s – a)} = e" S() (Ist Shifting Theorem) 15 cosh at L{{(t – a)u(t – a)} = e~**F(s) t-Shifting at sin wt 16 (8 – a)² + w² {e"F(s)} = f(t – a)u(t – a) (2nd Shifting Theorem) 8-a 17 eat cos wt (S • 9)(t) = [(r)a(t – r)dr (6 – a)² + w² Convolution 1. (1 – cos wt) 18 s(s² + w²) CJ • 9) - C)C() 19 (wt – sin wt) 1 (sin wt – wt cos wt) 2 1 20 (8² + w?)Z sin wt 21 (s² + w?)Z 1 (sin wt + wt coswt) 22 (s² + wa )Z (a² # b³) 1 (cos at - cos bt) 23 (s² + a²)(s² + b²) 62 - a 24 u(t – a) 25 eas 6(t – a)
The charge on a capacitor q(t) in a circuit with a resistor, a capacitor and an inductor connected in series driven by a given time-dependent voltage v(t) is governed by the second-order ODE d"(t) + 2q'(t) + 2q(t) = v(t). (1) A schematic of the RLC circuit can be found in the course notes. In Questions 1 - 6 you will calculate Laplace transforms / inverse Laplace transforms of func- tions. These calculations will be useful in determining the response of (1) for an initial value problem specified in Question 7. You MUST use the method of solution as per the instructions found in each of the below questions. 1 1. Use the linearity of property of Laplace transforms and the table of Laplace transforms on MyUni to find the Laplace transform of v(t) = av1 (t) + Bv2(t), where vi(t) = 6(t – 2) and v2(t) = u(t – 3). Here, a and ß are constants, and 6(t) and u(t) are the Dirac delta function and unit step function. 2 2. Complete the square of the denominator and use the s-shifting theorem to find the inverse Laplace transform of the function F(s) = 82 + 28 + 2° Where applicable, you may use entries (1) through to (15) and the basic general formulae from the table of Laplace transforms on MyUni. You must NOT use any entries (16) through to (23). 5 3. Use partial fraction decomposition, linearity and s-shifting theorem to find the inverse Laplace transform of the function Н() — s(s² + 2s + 2)* Where applicable, you may use entries (1) through to (15) and the basic general formulae from the table of Laplace transforms on MyUni. You must NOT use any entries (16) through to (23).
Transcribed Image Text:The charge on a capacitor q(t) in a circuit with a resistor, a capacitor and an inductor connected in series driven by a given time-dependent voltage v(t) is governed by the second-order ODE d"(t) + 2q'(t) + 2q(t) = v(t). (1) A schematic of the RLC circuit can be found in the course notes. In Questions 1 - 6 you will calculate Laplace transforms / inverse Laplace transforms of func- tions. These calculations will be useful in determining the response of (1) for an initial value problem specified in Question 7. You MUST use the method of solution as per the instructions found in each of the below questions. 1 1. Use the linearity of property of Laplace transforms and the table of Laplace transforms on MyUni to find the Laplace transform of v(t) = av1 (t) + Bv2(t), where vi(t) = 6(t – 2) and v2(t) = u(t – 3). Here, a and ß are constants, and 6(t) and u(t) are the Dirac delta function and unit step function. 2 2. Complete the square of the denominator and use the s-shifting theorem to find the inverse Laplace transform of the function F(s) = 82 + 28 + 2° Where applicable, you may use entries (1) through to (15) and the basic general formulae from the table of Laplace transforms on MyUni. You must NOT use any entries (16) through to (23). 5 3. Use partial fraction decomposition, linearity and s-shifting theorem to find the inverse Laplace transform of the function Н() — s(s² + 2s + 2)* Where applicable, you may use entries (1) through to (15) and the basic general formulae from the table of Laplace transforms on MyUni. You must NOT use any entries (16) through to (23).
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