By Peter V. O'Neil

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2y2 − 9xy + 3xy − 6x2 y = 0 (Hint: try xa y b ) 11. 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.

Hint: Solve for the amount of salt in tank 1 at time t first and then use this solution to determine the amount in tank 2. 5 gal/min 12. x2 − 2x y + x2 − 5x + 4 y = x4 − 2x3 e−x y 3 = 18e−3 1 2 lb/gal Tank 1 13. y − y = 2e4x y 0 = −3 14. y + 5y = 3x3 + x y −1 = 4 9x Tank 2 5 gal/min 15. Find all functions with the property that the y-intercept of the tangent to the graph at x y is 2x2 . 12 Mixing between tanks in Problem 17. Exact Differential Equations We continue the theme of identifying certain kinds of first-order differential equations for which there is a method leading to a solution.