By Peter V. O'Neil
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Additional resources for Advanced Engineering Mathematics, 6th Edition
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. We can write any first order equation y = f x y in the form M x y +N x y y = 0 For example, put M x y = −f x y and N x y = 1. 4 Exact Differential Equations 27 In this event, the differential equation becomes x dy =0 y dx + which, by the chain rule, is the same as d dx =0 x yx But this means that x yx =C with C constant. If we now read this argument from the last line back to the first, the conclusion is that the equation x y =C implicitly defines a function y x that is the general solution of the differential equation.
How much salt is in the tank when it contains 100 gallons of brine? Hint: The amount of brine in the tank at time t is 50 + t. 17. 12. Tank 1 initially contains 20 pounds of salt dissolved in 100 gallons of brine, while tank 2 contains 150 gallons of brine in which 90 pounds of salt are dissolved. At time zero a brine solution containing 21 pound of salt per gallon is added to tank 1 at the rate of 5 gallons per minute. Tank 1 has an output that discharges brine into tank 2 at the rate of 5 gallons per minute, and tank 2 also has an output of 5 gallons per minute.
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. How does the general solution of M + Ny = 0 obtained by using differ from that obtained using + c?
Advanced Engineering Mathematics, 6th Edition by Peter V. O'Neil