Question

Differential Equations Need Help! Will Rate!

Question 1 (35) 1. Build the characteristic polynomial for the DE z',-4x,-52-0 and find two particular solutions. Here, x' = dx/dt, x" = d2x/dt2. (15) 2. Verify that the two solutions are linearly independent. (5) 3. Build the general solution to the DE as a linear combination of these two solutions. (5) 4. Using the general solution, calculate the solution for the same DE with the initial conditions z(0) 5, x(0) 3. (10) Question 2 (30) 1. Find a solution to the initial value problem dy 2y + 2, y(3)=5 Make sure you solve all the way to the end i.e. express the solution in the form y f(t). There are points for this. (20) 2. Is there a guarantee that this solution is unique in the neighborhood of t 3? Why or why not? (10) Question 3 (35) 1. Consider the DE dx dt Calculate the fixed points of this DE. (10) 2. Where are the bifurcation points of this DE? (10) 3. Make qualitative sketches of f(x) vs z which capture the features of late time behavior of the DE for three well-chosen values of k. (10) 4. Sketch the bifurcation diagram for this DE. (5)

Answer 1

According to the rules only first four subparts to be answered

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