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====== Math 445 Lab 11: Differential Equations ====== ===== Problem 1: dy/dt = sin(y+t) ===== Show a quiver plot in the //(t,y)// plane of the differential equation //dy/dt = sin(y+t)// over the region //0 ≤ t ≤ 2pi// and //-pi ≤ y ≤ pi//. Use Matlab's //ode45// function to compute solution curves //y(t)// for each of the following initial conditions, and superimposed the solution curves on the quiver plot in the indicated colors //y(0) = -1// in red, //y(0) = 0// in blue , //y(0) = 1// in black. Be sure to set the axes tight around the quiver plot, set the unit length of the axes to be equal, label the axes, and title the graph. ===== Problem 2: van der Pol oscillator ===== ==== (a) ===== The van der Pol oscillator is defined by the second-order differential equation <latex> d^2x/dt^2 - \mu (1-x^2) dx/dt + x = 0 </latex> This second-order equation can be converted to a system of two first-order equations by letting //y// be a 2-d vector $y = [y_1, y_2] = [x, ~dx/dt]$. Then $dy/dt = [dy_1/dt, ~dy_2/dt ] = [y_2, ~\mu (1-y_1^2) y_2 - y_1]$. Use this definition of the vector $dy/dt$ to produce a quiver plot of the van der Pol oscillator in the $y_1, y_2$ plane for $\mu = 1$, over the range $-3 \leq y_1 \leq 3$ and $-3 \leq y_2 \leq 3$. Using Matlab's //ode45//, compute solution curves $y(t)$ for the five different initial values $y(0) = [0.1~ 0], [0.2~0], [1~0], [2~0],$ and $[3~0]$, and superimpose them on the quiver plot in red. Hint: put the call to //ode45// inside a //for// loop that sets the value of $y_1$ to the five different initial values. ==== (b) ==== Put your work for part (a) in a script file, and run the script for a range of values of $\mu$ between -1 and 1. How does the character of the solutions of the equation change as $\mu$ passes through 0? Turn in plots for a negative, zero, and a positive value of $\mu$, whichever values you think best illustrate the change as $\mu$ passes through zero. Hint: To avoid the need to change the value of $\mu$ in both the script and the function file that defines $dy/dt = f(t,y)$, define $dy/dt = f(t,y)$ as an anonymous function within your script file, so that it inherits the $\mu$ of the script file. ===== Problem 3: projectile motion ===== It is easy to determine the trajectory of a projectile subject to gravity if you neglect air resistance. using elementary physics and calculus, you can show that a projectile will travel farthest across a flat surface if you shoot it at an angle $\theta = \pi/4$ above the horizontal. In this problem you will determine the optimal angle and the maximum range of a projectile subject to air resistance. Use the following equations of motion, developed during class <latex> dx/dt = v_x \\ dy/dt = v_y \\ dv_x/dt = -mu v_x \sqrt{v_x^2 + v_y^2} \\ dv_y/dt = -g - mu v_x \sqrt{v_x^2 + v_y^2} </latex>
