Clear work and no blurry pictures. Please show the code clearly.

Problem #1 (35%) Consider the following first order dynamic system: 7* + 21x = 36 ut) a) Find the transfer function from U(s) to Y(s). b) What is the system time constant? What is the DC gain (i.e. static sensitivity)? c) Assume that input u(t) consists of a step function and a ramp function, such that u(t) = 6 + 2t. Without solving for y(t), qualitatively describe the steady-state response you expect from this linear time-invariant (LTI) system. d) Based on the input from part (c), and assuming zero initial conditions, solve for y(t). e) State whether the response of part (d) consists of a free response, forced response, or both. Explain your reasoning in 1-2 sentences. f) Using MATLAB and your transfer function from part (a), simulate the system response over the time interval from 0 to 10 seconds. In other words, use the tf and lsim commands to generate a time-domain system response. Be sure to add axis labels to your plot. Include your code for this plot. g) Numerically compute y(t) from 0 to 10 seconds based on your answer from part (d). Using the hold on function, superimpose a plot of y(t) on top of your plot from part (f). Include your code and the plot showing responses from parts (f) and (g). [Hint: Your responses should be identical! Adjust the plot settings so the two responses can be individually identified. Add a legend to distinguish the responses as coming from either the lsim simulation or your y(t) equation.] h) Do your response plots agree with your expectation from part (c)? Explain. Problem #1 (35%) Consider the following first order dynamic system: 7* + 21x = 36 ut) a) Find the transfer function from U(s) to Y(s). b) What is the system time constant? What is the DC gain (i.e. static sensitivity)? c) Assume that input u(t) consists of a step function and a ramp function, such that u(t) = 6 + 2t. Without solving for y(t), qualitatively describe the steady-state response you expect from this linear time-invariant (LTI) system. d) Based on the input from part (c), and assuming zero initial conditions, solve for y(t). e) State whether the response of part (d) consists of a free response, forced response, or both. Explain your reasoning in 1-2 sentences. f) Using MATLAB and your transfer function from part (a), simulate the system response over the time interval from 0 to 10 seconds. In other words, use the tf and lsim commands to generate a time-domain system response. Be sure to add axis labels to your plot. Include your code for this plot. g) Numerically compute y(t) from 0 to 10 seconds based on your answer from part (d). Using the hold on function, superimpose a plot of y(t) on top of your plot from part (f). Include your code and the plot showing responses from parts (f) and (g). [Hint: Your responses should be identical! Adjust the plot settings so the two responses can be individually identified. Add a legend to distinguish the responses as coming from either the lsim simulation or your y(t) equation.] h) Do your response plots agree with your expectation from part (c)? Explain