Download Problems in Thermodynamics, Ordinary Differential Equations, and Linear Algebra - Prof. Tr and more Exams Differential Equations in PDF only on Docsity! Samples for Final Problems ♯ 1. A bottle of beer at 50oF is discovered in a kitchen counter in a room of 70oF . Ten minutes later, the bottle is at 60oF . If the refrigerator is kept at 40oF , how long had the bottle been sitting on the counter when it was first discovered? Assume the temperature satisfies Newton’s law of cooling. (Similar problems may deal with radioactive decay, savings accounts or loans.) ♯ 2. Consider the autonomous ODE dx dt = −(x + 1)(x2 − 9). (a) Sketch the phase line diagram. Your sketch should include the equi- librium points and show the directions of motion. Identify the stability of each equilibrium point. (b) Use the information of (a) to sketch the solution curves in the (t, x)– plane (t ≥ 0) for the initial values x(0) = 0, 3, 4. ♯ 5. True of False? (a) The set S = {[t, s]T | t2 + s2 = 1} in R2 is the solution set of a system of linear equations. (b) The set S = {[1, t, s]T | t, s ∈ R} in R3 is a subspace of R3. (c) The set S = {[1, t, s]T | t, s ∈ R} in R3 is the solution set of a system of linear equations. ♯ 6. Find the value of the parameter a for which the system of equations x + y + z = 1 x + 2y + 2z = 2 y + z = a has a solution. For that value, determine a parametric representation of the solution set. Use this to find a basis of the nullspace of the matrix of coefficients on the right hand sides. ♯ 7. Are the vectors v1 = 1 2 1 , v2 = 1 1 0 , v3 = 1 −1 −2 linearly dependent or linearly independent? If they are linearly dependent, find numbers c1, c2, c3, not all zero, such that c1v1 + c2v2 + c3v3 = 0 ♯ 8. Find the solution to the initial value problem x′ = −1 2 −2 −1 x, x(0) = 1 0 . ♯ 9. Find a fundamental set of solutions of x′ = Ax for A = 1 1 0 1 0 1 0 1 1 . ♯ 10. Sketch the phase plane portrait of the system x′ = Ax for the following matrices A: (a) A = −1 2 −2 −1 , (b) A = 1 0 2 3 . Your sketch should include all half-line solutions (if exist) and identify slow and fast motions in case of nodes, or show the correct direction of rotation in case of spirals or centers.
