Uh need to solve only (question 3) I also attached the table that is given in questions to solve uh only need to solve the q3 in one hour and get thumb up plz plz plz solve perfectly and handwritien soloution needed plz I need perfect soloution according to question requirements plz The charge on a capacitor q(t) in a circuit with a resistor, a capacitor and an inductor connectedin series driven by a given time-dependent voltage v(t) is governed by the second-order ODEd"(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 valueproblem specified in Question 7.You MUST use the method of solution as per the instructions found in each of the belowquestions.1 1. Use the linearity of property of Laplace transforms and the table of Laplace transforms onMyUni to find the Laplace transform ofv(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 inverseLaplace transform of the functionF(s) =82 + 28 + 2°Where applicable, you may use entries (1) through to (15) and the basic general formulaefrom 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 inverseLaplace transform of the functionН() —s(s² + 2s + 2)*Where applicable, you may use entries (1) through to (15) and the basic general formulaefrom the table of Laplace transforms on MyUni. You must NOT use any entries (16)through to (23). | 25 |**| OLt - a)1:41:4Table of Laplace TransformsF(s) = L{S(t)}f(t)1/s1/s²1/s" (n = 1,2, ...)| 1/V1/s3/21/s (a 0)11t"-1/(n – 1)!1//mt34Basic General Formulas for the Laplace Transform6.t0-1/T(a)1FormulaName, Commentseat8- a1F(s) = L{S(1} = [e"()dtDefinition of Transformteat(8 – a)²1| (1) = c-'{F(»)}Inverse Transform9.(8 – a)™1(n = 1, 2, .)(n-1)!1(eat – e")C{af(t) + bg(t)} = aL{f(t)} + bL{g(t)}Linearity1(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)12sin wts² + w?cism) = "L(J) – *-'s(0) – .. – fln-1)(0)13Cos wtC{e“{(t)} = F(s – a)-Shiftinga14sinh atL-{F(s – a)} = e" S()(Ist Shifting Theorem)15cosh atL{{(t – a)u(t – a)} = e~**F(s)t-Shiftingat sin wt16(8 – a)² + w²{e"F(s)} = f(t – a)u(t – a)(2nd Shifting Theorem)8-a17eat cos wt(S • 9)(t) =[(r)a(t – r)dr(6 – a)² + w²Convolution1.(1 – cos wt)18s(s² + w²)CJ • 9) - C)C()19(wt – sin wt)1(sin wt – wt cos wt)2120(8² + w?)Zsin wt21(s² + w?)Z1(sin wt + wt coswt)22(s² + wa )Z(a² # b³)1(cos at - cos bt)23(s² + a²)(s² + b²)62 - a24u(t – a)25eas6(t – a)

Question

Uh need to solve only (question 3) I also attached the table that is given in questions to solve uh only need to solve the q3 in one hour and get thumb up plz plz plz solve perfectly and handwritien soloution needed plz I need perfect soloution according to question requirements plz The charge on a capacitor q(t) in a circuit with a resistor, a capacitor and an inductor connectedin series driven by a given time-dependent voltage v(t) is governed by the second-order ODEd&#34;(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 valueproblem specified in Question 7.You MUST use the method of solution as per the instructions found in each of the belowquestions.1 1. Use the linearity of property of Laplace transforms and the table of Laplace transforms onMyUni to find the Laplace transform ofv(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 inverseLaplace transform of the functionF(s) =82 + 28 + 2°Where applicable, you may use entries (1) through to (15) and the basic general formulaefrom 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 inverseLaplace transform of the functionН() —s(s² + 2s + 2)*Where applicable, you may use entries (1) through to (15) and the basic general formulaefrom the table of Laplace transforms on MyUni. You must NOT use any entries (16)through to (23). | 25 |**| OLt - a)1:41:4Table of Laplace TransformsF(s) = L{S(t)}f(t)1/s1/s²1/s&#34; (n = 1,2, ...)| 1/V1/s3/21/s (a > 0)11t&#34;-1/(n – 1)!1//mt34Basic General Formulas for the Laplace Transform6.t0-1/T(a)1FormulaName, Commentseat8- a1F(s) = L{S(1} = [e&#34;()dtDefinition of Transformteat(8 – a)²1| (1) = c-'{F(»)}Inverse Transform9.(8 – a)™1(n = 1, 2, .)(n-1)!1(eat – e&#34;)C{af(t) + bg(t)} = aL{f(t)} + bL{g(t)}Linearity1(a # b)10(8 – a)(s – b)(a – b)1(aet – be&#34;)- sCf) – f(0)(a ± b)11(8 – a(s – b)(a – b)Differentiation of Function- PL(S) – sf(0) – s'(0)12sin wts² + w?cism) = &#34;L(J) – *-'s(0) – .. – fln-1)(0)13Cos wtC{e“{(t)} = F(s – a)-Shiftinga14sinh atL-{F(s – a)} = e&#34; S()(Ist Shifting Theorem)15cosh atL{{(t – a)u(t – a)} = e~**F(s)t-Shiftingat sin wt16(8 – a)² + w²{e&#34;F(s)} = f(t – a)u(t – a)(2nd Shifting Theorem)8-a17eat cos wt(S • 9)(t) =[(r)a(t – r)dr(6 – a)² + w²Convolution1.(1 – cos wt)18s(s² + w²)CJ • 9) - C)C()19(wt – sin wt)1(sin wt – wt cos wt)2120(8² + w?)Zsin wt21(s² + w?)Z1(sin wt + wt coswt)22(s² + wa )Z(a² # b³)1(cos at - cos bt)23(s² + a²)(s² + b²)62 - a24u(t – a)25eas6(t – a)

Accepted Answer

To find the inverse laplace transform of given function H(s) first we will take the partial fraction…

5 3. Use partial fraction decomposition, linearity and s-shifting theorem to find the inverse Laplace transform of the function H(s) s(s² + 2s + 2)'