By Brannan J., Boyce W.
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Extra resources for Differential equations. An introduction to modern methods and applications
Hint: Consider the general solution, Eq. (42), and use L’Hospital’s rule on the first term. 33. Show that if a and λ are positive constants, and b is any real number, then every solution of the equation y + ay = be−λt has the property that y → 0 as t → ∞. Hint: Consider the cases a = λ and a = λ separately. In each of Problems 34 through 37 construct a first order linear differential equation whose solutions have the required behavior as t → ∞. Then solve your equation and confirm that the solutions do indeed have the specified property.
N. 13 2 3 4 t y y y y = −2 + t − y = te−2t − 2y = e−t + y = t + 2y n−1 u(tn ) = (1 − k t)n u 0 + kT0 t (1 − k t) j , j=0 Direction field for Problem 20. In each of Problems 21 through 28 draw a direction field for the given differential equation. Based on the direction field, determine the behavior of y as t → ∞. If this behavior depends on the initial value of y at t = 0, describe this dependency. Note that the right sides of these equations depend on t as well as y. 21. 22. 23. 24. and so forth.
14. 13. In each of Problems 15 through 20 identify the differential equation that corresponds to the given direction field. (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) y y y y y y y y y y = 2y − 1 =2+y = y−2 = y(y + 3) = y(y − 3) = 1 + 2y = −2 − y = y(3 − y) = 1 − 2y =2−y 15. 16. 17. 18. 19. 20. 8 2 3 4 t Direction field for Problem 15. 9 2 3 4 t Direction field for Problem 16. 10 Direction field for Problem 17. 1 Mathematical Models, Solutions, and Direction Fields y 1 2 3 4 t –1 –2 –3 25. 26. 27.
Differential equations. An introduction to modern methods and applications by Brannan J., Boyce W.