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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 $y = [y_1, y_2] = [x, ~dx/dt]$. Then $dy/dt = [y_2, ~\mu (1-y_1^2) y_2 + y_1]$. 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 $-3 \leq y_2 \leq 3$. Compute solution curves
