Question:

Solve d²y/dt²= -1/y³ subject to y(0) = yo and y'(0) = 0.

Solve d²y/dt²= -1/y³ subject to y(0) = yo and y'(0) = 0.

Solve d²y/dt²= -1/y³ subject to y(0) = yo and y'(0) = 0.

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C.
Post + 2xy +"(9)
However, there is no function h(y) that would make this equal to N = x +2y, for were
this the case,
11.
+2xy + '(y) = x + xy,
which implies
W'(y) = -x -xy
But this is impossible since the right-hand side depends on x and is not a functi

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Differential equations Solutions

C. Post + 2xy +"(9) However, there is no function h(y) that would make this equal to N = x +2y, for were this the case, 11. +2xy + '(y) = x + xy, which implies W'(y) = -x -xy But this is impossible since the right-hand side depends on x and is not a function of y alone. We will see how to solve an equation such as this in Section 3.5. . EXERCISES 3.3 13. 2y sin(xy) dx + (2x sin(xy) + 3y) dy = 0, y(0) = 1 z ? dx = 0, y(O)=1 In Exercises 1-10, determine if the differential equation is exact. If it is exact, find its solution. 1. (3x2 - 4y2) dx - (8xy - 12y3) dy = 0 2. (3xy + 4y2) dx + (5x2y + 2x2) dy = 0 3. (2xy + ye') dx + (x2 +et) dy = 0 4. (2xe* + x?yety - 2) dx +xety dy = 0 5. (2x cos y - x) dx + x2 sin y dy = 0 6. (y cos x + 3et cos y) dx +(sin x-3e" sin y) dy = 0 1-2xy x2 - 2xy + 1 7. y'= 8. y' = r2 x² - y² ds el - 2t coss dr -r cos e 10 9. 10. di es - 12 sins de r + sin e n Exercises 11-14, find the solution of the initial value roblem. 1. 2xy dx + (x2 + 3y2) dy = 0, y(1) = 1 2xel - 3x2y 2. y' x3 – x²e" -, y(1)=0 14. 1 dyt x² + y² x2 + y2 15. Use Maple (or another appropriate software pack- age) to graph the solution in Exercise 11. 16. Use Maple (or another appropriate software pack- age) to graph the solution in Exercise 12. 17. Show a separable differential equation is exact. 18. Show the converse of Theorem 3.2 can be proved by integrating with respect to y first. y 19. Determine conditions on a, b, c, and d so that the differential equation ax + by cx tdy is exact and, for a differential equation satisfying these conditions, solve the differential equation. y exactress + N(x, y) I equation 0.