Uh need to solve only (question 8) solve the q8 in 30 min and get thumb up plz plz plz solve perfectly and handwritien soloution needed plz I also attached the answers of q 7( value of q(t)) for ur help I need perfect soloution according to question requirements plz - Lt-2)de sim(t-2) + P- (t-3)-Ie cos (t-3)a(t) =%3D2(t-3)esinct-3) The charge on a capacitor g(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 ODE"(t) + 24'(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 fune-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.I 1. Use the linearity of property of Laplace transforms and the table of Laplace transforms onMyUni to find the Laplace transform ofv(t) = avy (t) + Buy (t),where (t) = 6(t-2) and ty(t) = u(t – 3). Here, a and 3 are constants, and d(t) and u(t)are the Dirac delta function and unit step function.1 2. Complete the square of the denominator and use the s-shifting theorem to find the inverseLaplace transform of the functionF(e) = 7 +2s+2Where 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).3. Use partinl fraction decomposition, linearity and s-shifting theorem to find the inverseLaplace transform of the function1H(s) =s(s + 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).4. Use the convolution theorem and your answer to Question 2 to verify the inverse Laplacetransform of the function H(s) found in Question 3. i.e. Find the inverse Laplace transformof the functionH(s) =s(8² + 2s+ 2)F().%3DYou MUST evaluate the resulting integral by hand, and clearly show all your working.5. Use the t-shifting theorem and your answer to Question 2 to find the inverse Laplace trans-form of the functione-2K(s) =2+2s+2°Use the t-shifting theorem and your answer to Question 3 to find the inverse Laplace trans-form of the functionJ(s) =s(s² +2s +2)'7. Use Laplace transforms and the results from Questions 1 -6 to find the response of (1)subject to the initial conditionsq(0) =0 and d(0) = 1,with the time-dependent voltagev(t) = ad(t – 2) + Bu(t - 3).8. Plot the response, q(t), found in Question 7, on the interval 0

Question

Uh need to solve only (question 8)  solve the q8 in  30 min and get thumb up plz plz plz solve perfectly and handwritien soloution needed plz I also attached the answers of q 7( value of q(t))  for ur help I need perfect soloution according to question requirements plz - Lt-2)de sim(t-2) + P- (t-3)-Ie cos (t-3)a(t) =%3D2(t-3)esinct-3) The charge on a capacitor g(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 ODE&#34;(t) + 24'(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 fune-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.I 1. Use the linearity of property of Laplace transforms and the table of Laplace transforms onMyUni to find the Laplace transform ofv(t) = avy (t) + Buy (t),where (t) = 6(t-2) and ty(t) = u(t – 3). Here, a and 3 are constants, and d(t) and u(t)are the Dirac delta function and unit step function.1 2. Complete the square of the denominator and use the s-shifting theorem to find the inverseLaplace transform of the functionF(e) = 7 +2s+2Where 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).3. Use partinl fraction decomposition, linearity and s-shifting theorem to find the inverseLaplace transform of the function1H(s) =s(s + 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).4. Use the convolution theorem and your answer to Question 2 to verify the inverse Laplacetransform of the function H(s) found in Question 3. i.e. Find the inverse Laplace transformof the functionH(s) =s(8² + 2s+ 2)F().%3DYou MUST evaluate the resulting integral by hand, and clearly show all your working.5. Use the t-shifting theorem and your answer to Question 2 to find the inverse Laplace trans-form of the functione-2K(s) =2+2s+2°Use the t-shifting theorem and your answer to Question 3 to find the inverse Laplace trans-form of the functionJ(s) =s(s² +2s +2)'7. Use Laplace transforms and the results from Questions 1 -6 to find the response of (1)subject to the initial conditionsq(0) =0 and d(0) = 1,with the time-dependent voltagev(t) = ad(t – 2) + Bu(t - 3).8. Plot the response, q(t), found in Question 7, on the interval 0

Accepted Answer

Given : response 
         qt=αe-t-2sint-2+β12-12e-t-3cost-3-12e-t-3sint-3
We have to plot the…