By Peter V. O'Neil
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Extra info for Advanced Engineering Mathematics
Y' - Y = 2e4x ; Y(0 ) = - 3 14. y' + 9z = 3x 3 + x ; y(-1) = 4 15. Find all functions with the property that the y-intercept of the tangent to the graph at (x, y) is 2x 2 . 12 Mixing between tanks in Problem 17. Exact Differential Equation s We continue the theme of identifying certain kinds of first-order differential equations for whic h 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, a function CO such that ax = y) and N(x, y) = M(x, y) and 1.
Integrate to get xy = 3 4x4 +C . Then y (x) = 3 3 4x C +x for x > 0 . For the initial condition, we need 4+ C y(1) =5= so C = 17/4 and the solution of the initial value problem i s 3 17 y(x) = 4x + 4x 3 forx>0 . 6 in closed form (as a finite algebraic combination of elementary functions) . Thi s occurs with y' +xy=2 . whose general solution is y(x) = 2e-x2/2 f ex2/2 dx+ Ce -x2/2 . We cannot write f e x2/2 dx in elementary terms . 10. This provides some idea of the behavior of solutions, at least within the range of the diagram .
3y4 - 1 + 12xy3 Y' = 0; Y( l ) = 2 10. 2y - y2 sec2 (xy 2) + (2x - 2xysec 2 (xy 2))y' = 0; Y( l ) = 2 11. xcos(2y - x) - sin(2y - x) - 2xcos(2y - x)y'=0 ; y(7r/12) = *r/ 8 y(4) = 4 14. eY + (xeY -1)y' = 0 ; y(5) = 0 0 In each of Problems 9 through 14, determine if the differential equation is exact in some rectangle containing in it s interior the point where the initial condition is given . If so, solve the initial value problem . This solution may b e implicitly defined . If the differential equation is not exact, do not attempt a solution .
Advanced Engineering Mathematics by Peter V. O'Neil