#3 MATH 310: Ordinary Differential Equations Due date: Friday October 14, 2016 1. Use Euler's method to solve the nonlinear ODE problem as described in exercise #15 on page 111. To compute your numerical solution and plot the results, you have a few options: Modify one of the codes that I posted on Canvas for Maple (euler.mw) or Matlab (euler.m). Write your own code using a language of your choice. Regardless of which approach you choose, provide a print-out of your code. Submit a single plot that displays your solutions from parts (a) and (b) on the same set of axes so that they can be easily compared. 2. Solve the following initial value problems: (a) (b) 4y 4y y = 0; 4y 4y + y = 0; y(0) = 31 , y (0) = 0. y(0) = 31 , y (0) = 0. y (0) = 2. (c) y + 2y + 4y = 0; y(0) = 1, (d) 2y y = 0; y(2) = 1, y (2) = 2. In each part, provide a plot of your solution and indicate the behaviour as t increases. 3. For each case below, write down a linear, constant-coefficient, homogeneous ODE of lowest order for which the given expression is a solution: (c) 7ex cos (x) (a) e5x e2x (b) e 2t + 2e 2t 4. (a) Verify that y1 (t) = t2 and y2 (t) = t1 are two solutions of the differential equation t2 y 2y = 0 for t > 0. Then, use the theorems we discussed in class to show that any solution of the ODE (for t > 0) can be written in the form y = c1 t2 + c2 t1 , where c1 and c2 are arbitrary constants. Carefully justify your answer. (b) Verify that y1 (t) = 1 and y2 (t) = t1/2 are solutions of the ODE yy + (y )2 = 0 for values of t > 0. Then show that y = c1 + c2 t1/2 is not, in general, a solution of this equation. Explain why this result does not contradict Theorem 3.2.2 (the superposition principle). 5. Exercise #21, page 144. 6. Exercise #28, page 156. 7. If the Wronskian of f (x) and g(x) is W (x) = ex , and if f (x) = e2x , then find g(x). 1