MAT B44 Assignment 2 Jordan Bell Department of Mathematics, University of Toronto Due in class, Tuesday October 25. This is the same day as the midterm which will be held in class. R 1 In this assignment if there is a pointR where you have to integrate t dt, you R can assume we are using t > 0 so that 1t dt = ln t instead of 1t dt = ln |t|. 1. (5 marks) Figure 1 plots the integral curves for one of the following five ODE: (i) y 0 = t2 + t2 . 2 (ii) y 0 = y/t + yt 2 . (iii) y 0 = ty. (iv) y 0 = y + yt2 . 2 (v) y 0 = yt 2 . Argue why four of these cannot be the differential equation the figure corresponds to. 2. (5 marks) For IVP ty 0 (t) = t2 + y(t), y(1) = 0, approximate y(1.03) using Euler's method with step size h = 0.01. 3. (5 marks) Calculate the Picard iterates 0 , 1 , 2 for the IVP \u0012 \u0013 y(t) 0 y (t) = r 1 y(t), r > 0, K > 0, y(0) = y0 . K Part of the skill being developed here is being comfortable with complicated calculations. 2 (t) is ugly to work out but is reasonable. 3 (t) would be too long to ask you for this ODE. 4. (5 marks) For an annual interest rate of r = 0.04 and for an annual payment of A = 1000, let y0 be an initial debt. After n years, the debt is yn . Write this as a difference equation. Determine y0 so that after 20 years the debt is paid off, that is y20 = 0. 5. (5 marks) Find a general solution of y 00 4y 0 2y = 0. Find solution for IVP y(0) = 0, y 0 (0) = 1. 6. (5 marks) Find a general solution of y 00 4y 0 + 5y = 0. Find solution for IVP y(0) = 0, y 0 (0) = 1. 1 Figure 1: Integral curves 2 7. (5 marks) Find a general solution of 4y 00 + 20y 0 + 25y = 0. Calculate the Wronskian of the differential equation using (i) the general solution and (ii) Abel's formula. (The Wronskian is determined up to a constant multiple, so for example you might get W = t calculating it one way and W = 2t another way, but you won't get W = t and W = t2 . Sometimes it matters which way you calculate it, and when it does I emphasize this by writing W (y1 , y2 ) to show that the constant depends on y1 , y2 .) 8. (5 marks) Find a general solution of 1 y 00 (t) y 0 (t) = t t using the method of substitution z(t) = y 0 (t). 9. (5 marks) Let L[y] = y 00 + py 0 + qy. If L[y1 ] = 0 and L[y2 ] = 0, let W be the Wronskian of y1 , y2 . Show that \u0012 \u00130 W y2 = 2. y1 y1 \u0001 Let L[y] = y 00 + x1 y 0 + 1 4x1 2 y. Use x > 0. First, use Abel's formula to calculate the Wronskian of the ODE; this is determined up to a constant but that will not matter. Given that y1 (x) = x1/2 sin x is a solution, use the above formula involving the Wronskian to calculate a solution y2 . 10. (10 marks) Find a general solution of the inhomogeneous ODE y 00 3y 0 + 5y = 1 using variation of parameters. Be careful doing the calculations. It is easy to miss a negative sign and because this is an assignment instead of a midterm, making a calculation mistake will not be forgiven because you have time to double check your work. (In fact finding a solution to the inhomogeneous equation is easier to do here using undetermined coefficients, but the point of the question is to practice variation of parameters.) 3