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====== Math 445 Lab 11: Differential Equations ====== ===== Problem 1 ===== 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 ===== 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> Convert this second-order equation to a system of two first-order equations by letting //y// be a 2-d vector with <latex> y_1 = x<\latex> and <latex>y_2 = dx/dt</latex>. Then <latex> dy_1/dt = y_2</latex> and <latex> dy_2/dt = \mu (1-y_1^2) y_2 + y_1</latex>
