Part A and E **Mass-Spring System with Damping: Examining Motion Equations**The motion of a mass-spring system with damping is described by the differential equation:\[ y''(t) + b y'(t) + 9y(t) = 0 \] with initial conditions \( y(0) = 1 \) and \( y'(0) = 0 \).Determine and sketch the equations of motion for different values of the damping coefficient \( b \): 0, 2, 6, and 10.---**Equation of Motion for Different \( b \) Values:**- **For \( b = 0 \):** \[ y(t) = \_\_ \] *(Type the exact answer, using radicals as needed.)*- **For \( b = 2 \):** \[ y(t) = \_\_ \] *(Type the exact answer, using radicals as needed.)*- **For \( b = 6 \):** \[ y(t) = \_\_ \] *(Type the exact answer, using radicals as needed.)*- **For \( b = 10 \):** \[ y(t) = \_\_ \] *(Type the exact answer, using radicals as needed.)*---**Graph Selection:**Each graph represents the system's response for different \( b \) values:- **Graph A:** - \( b = 0 \) is a solid blue line. - \( b = 2 \) is a dotted red line. - \( b = 6 \) is a dashed green line. - \( b = 10 \) is a dash-dot black line.- **Graph B, C, and D:** (*Examine the key for line styles to determine which graph matches each description.*)Select the graph that correctly represents all solutions based on the indicated line styles for each \( b \) value: - **b = 0**: Solid blue line- **b = 2**: Dotted red line- **b = 6**: Dashed green line- **b = 10**: Dash-dot black lineUse this information to identify the correct graph that displays the motion equations for each damping coefficient value accurately.

Answered: Image /qna-images/answer/ec33ec64-8c04-4d77-ad9b-2c8fe83c57c2.jpg

The motion of a mass-spring system with damping is governed by y'' (t) + by' (t) +9y(t) = 0; y(0) = 1, and y'(0) = 0. Find the equation of motion and sketch its graph for b= 0, 2, 6, and 10. What is the equation of motion for b=0? y(t) = (Type an exact answer, using radicals as needed.) What is the equation of motion for b=2? y(t) = (Type an exact answer, using radicals as needed.) What is the equation of motion for b = 6? y(t) = (Type an exact answer, using radicals as needed.) What is the equation of motion for b = 10? y(t) = (Type an exact answer, using radicals as needed.) Choose the correct graph of the solutions. For each of the graphs b = 0 is a solid blue line, b = 2 is a dotted red line, b= 6 is a dashed green line, and b= 10 is a dash-dot black line. A. B. Q D.