MA-UA 262: Ordinary Differential Equations Summer 2020 Assignment 1 Due: Sunday July 19, by 11:59pm EST, via Gradescope • Office hours Friday, Saturday, Sunday from 11:30am-12:30am. Location https : //nyu.zoom.us/j/9179442616 • You can also email questions and I will try my best to help troubleshoot. Do not post questions on Campuswire. • Keep work within the scope of the class. Everything is doable using the techniques pre- sented in class. • Show all work. Skip out on steps and hand wave answers at your own risk. Remember nothing in this math business is free. • Failure to upload to Gradescope in time is akin to no submitted. As a result, a zero will be given. Failure to submit properly, i.e. assign pages to problems, will result in a heavy deduction of points. Don’t push your submission to the last minute!!! 1. (5 points) Find the general solution of y¨ + 2y˙ + y = 0 3. (5 points) Use the Euler approximations (linearized version i.e. section 1.13) to ap- proximate the solution to the IVP y˙ = t+ y y(0) = 1 at t = 3/2 using step size h = .5. 1 MA-UA 262: Ordinary Differential Equations Summer 2020 4. (10 points) A 40 liter tank is initially half full of water. A solution containing 10 grams per liter of salt begins to flow in at 4 liters per minute. The well mixed solution in the tank flows out of the tank at a rate of 2 liters per minute. How much salt is in the tank just before it overflows. 5. (5 points) Transform the equation t2y¨ + 3ty˙ + y = 2 t from the (t, y)-variables to the (s, v)-variables where v = y and s = ln t. Note that I am not asking you to solve the equation in the (t, y) or the (s, v) variables. Just transform the given equation. 6. (5 points) Transform the equation y˙ = tn−1f ( y tn ) from the (t, y)-variables to the (t, v) variables where v = y tn . Explain how you would solve it in the (t, v)-variables. 7. (10 points) Let a, b be constants. Show that the equation is exact 3t2 + 8at+ 2by2 + 3y + (4bty + 3t+ 5)y˙ = 0 and find the solution. 8. (10 points) Let a be a positive constant. Solve y¨ = a (1 + (y˙))1/2 Hint: Let w = y˙. Solve the first order equation in the w variable. Then solve for y. 9. Assume that y(t) is a function that satisfies y(t) = 1 + ∫ t 0 w(s) ds (1) where w(s) = − ∫ s 0 y(α) dα (a) (5 points) Show that y(t) defined in (1) satisfies the IVP y¨ + y = 0 y(0) = 1 y˙(0) = 0 2 MA-UA 262: Ordinary Differential Equations Summer 2020 (b) (10 points) Using the integral equation (1), we define the Picard sequence yn(t) = 1 + ∫ t 0 wn−1(s) ds where wn−1(s) = − ∫ s 0 yn−1(α) dα and n = 1, 2, 3, . . .. Compute y1, y2, y3, y4. Based on these computations, what is a formula for yn. 10. (10 points) Let’s find the general solution to the following ODE y˙ = y − t+ 2 2y + t+ 1 by following the outline described below. • Begin by converting the the ODE from the t, y variables to the s, v variables via the change of variables v = y − c1 and s = t− c2. • Now select c1, c2 so that the equation in the s, v variables is of the from dv ds = f (v s ) • Now solve for v. After which you can solve for y. Note that you may leave your solution as an implicit equation in the t, y variables. Remark: I am testing you on your ability to solve the equation using the method outlined above. If you want the credit for this problem, then follow the outline. 11. (10 points) Let a be a positive constant. For which values of a does the IVP y¨ + ay = 0 y(0) = 0 y(pi) = 1 have no solution. 12. (15 points) Assume that x(t) and w(t) are continuous non-negative functions on the interval [0, a]. Let C be a non-negative constant, and assume that w(t) ≤ C + ∫ t 0 x(s)w(s) ds, ∀t ∈ [0, a] (2) Show that w(t) ≤ C exp(F (t)) (3) where F (t) = ∫ t 0 x(s) ds Note that the constants C in (2), (3) are the same. 3
