7. Nаme this structure: (2 wоrds) 8. Nаme this structure: (3 wоrds) 9. Nаme this structure: (2 wоrds) of the bone
Prоblems 2 - 5 аre wоrth 15 pоints eаch. Problems 6 & 7 аre worth 10 points each. Points earned on problem 7 will be added to Exam 1 as a reassessment bonus opportunity. You must show all work that leads to your answer for full credit. Partial credit can be earned based on your responses. Problem 2 The parametric first-order system of differential equations d x d t = - 3 x - 13 y d y d t = 5 x - y has the phase plane below. Picture1.png (a) Identify the critical point (b) The critical point is considered a spiral point. Is this point stable or unstable? How do you know? (c) Sketch the curve for the initial condition x → ( 0 ) = 3 2 . There is no need to replicate the directional field. You can just provide a sketch of the curve in the x-y coordinate plane. Problem 3 Consider the vector functions x → 1 = e - t 4 e - t and x → 2 = - e - 6 t e - 6 t . (a) Use the Wronskian to show that these vectors are linearly independent for all t. (b) Show that the given vector functions are solutions to the homogeneous system x → ' = - 5 1 4 - 2 x → by finding the derivative vectors and plugging into the system. (Not by finding eigenvalues.) (c) Write the general form of the solution. (d) If x → ( 0 ) = 1 2 , find the particular solution curve. Problem 4 Find the general solution for the system x → ' = 1 2 3 2 x → . Problem 5 Picture2.png A two-tank system is modeled in the diagram above. Tank A initially contains 3 kg of salt dissolved in 24 L of water. Tank B initially contains 24 L of pure water. Pure water is pumped into Tank A at a rate of 6 liters per minute. The solution in Tank A is pumped into Tank B at a rate of 8 liters per minute. The solution in Tank B is pumped back into Tank A at a rate of 2 liters per minute. Additionally, a valve is taking out the solution in Tank B are a rate of 6 liters per minute. Find a complete solution for this system. Problem 6 Consider the first-order system of differential equations x → ' = 3 - 13 5 1 x → . (a) This system will have a complex conjugate pair of eigenvalues. Find the eigenvalues. (b) The corresponding eigenvectors for this system are given as u → = 1 ± 8 i 5 . The independent real vector solution for the system is x → = c 1 e α t cos β t a → - e α t sin β t b → + c 2 ( e α t sin β t a → + e α t cos β t b → ) . Write the general solution for this system. Problem 7 (a) Find the general solution for the equation d y d x = e x y 2 using separation of variables. (b) Find the general solution for the linear first-order differential equation x d y d x - 2 y = x 3 cos x .