By Peter V. O'Neil
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Extra info for Advanced Engineering Mathematics, International Student Edition
Y + y = y (Hint: try x y = x y =e y ) 12. x2 y + xy = −y−3/2 (Hint: try ax b x y = xa y b ) In each of Problems 13 through 20, find an integrating factor, use it to find the general solution of the differential equation, and then obtain the solution of the initial value problem. 6 2 = xa ebx ) y 0 = 7 /6 9. 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.
Integrate this equation to obtain 1 − = −e−x + k y an equation that implicitly defines the general solution. In this example we can explicitly solve for y, obtaining the general solution y= 1 −k e−x Now recall that we required that y = 0 in order to separate the variables by dividing by y2 . In fact, the zero function y x = 0 is a solution of y = y2 ex , although it cannot be obtained from the general solution by any choice of k. For this reason, y x = 0 is called a singular solution of this equation.
Also determine when the dog will eventually catch the man. (This is American Mathematical Monthly problem 3942, 1941). 26. (Pursuing Bugs) One bug is located at each corner of a square table of side length a. At a given time they begin moving at constant speed v, each pursuing its neighbor to the right. (a) Determine the curve of pursuit of each bug.
Advanced Engineering Mathematics, International Student Edition by Peter V. O'Neil