By Peter V. O'Neil

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X cos 2y − x − sin 2y − x − 2x cos 2y − x y = 0 y /12 = /8 y 12. 1 + ey/x − ey/x + ey/x y = 0 y 1 = −5 x 13. y sinh y − x − cosh y − x + y sinh y − x y = 0 y 4 =4 14. ey + xey − 1 y = 0 y 5 = 0 In Problems 15 and 16, choose a constant so that the differential equation is exact, then produce a potential function and obtain the general solution. 15. 2xy3 − 3y − 3x + x2 y2 − 2 y y = 0 16. 3x2 + xy − x2 y −1 y =0 17. Let be a potential function for M + Ny = 0 in some region R of the plane. Show that for any constant c, + c is also a potential function.

2xy2 + 2xy + x2 y + x2 y = 0 4 2 = ya ebx ) 20. 3x2 y + y3 + 2xy2 y = 0 y 2 = 1 21. Show that any nonzero constant multiple of an integrating factor for M + Ny = 0 is also an integrating factor. 22. Let x y be an integrating factor for M + Ny = 0 and suppose that the general solution is defined by x y = C. Show that x y G x y is also an integrating factor, for any differentiable function G of one variable. Homogeneous, Bernoulli, and Riccati Equations In this section we will consider three additional kinds of first-order differential equations for which techniques for finding solutions are available.

Examples involving pursuit problems are missiles fired at airplanes and a rendezvous of a shuttle with a space station. These are complex problems that require numerical approximation techniques. We will consider a simple pursuit problem that can be solved explicitly. Suppose a person jumps into a canal of constant width w and swims toward a fixed point directly opposite the point of entry into the canal. The person’s speed is v and the water current’s speed is s. Assume that, as the swimmer makes his way across, he always orients to point toward the target.

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