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1 FACULTAD DE AGRONOMIA CUARTA PRÁCTICA DIRIGIDA ANÁLISIS MATEMATICO PARA INGENIEROS INDUSTRIALES

ECUACIONES DIFERENCIALES EXACTAS I. Determinar si las ecuaciones diferenciales dadas son o no exactas.

16. (12𝑥 2 − 10𝑥𝑦 + 2𝑦 3 + 4)𝑑𝑥 + (6𝑥𝑦 2 − 5𝑥 2 )𝑑𝑦 = 0 17. (6𝑥 − 2𝑦 + 𝑦𝑒 𝑥𝑦 )𝑑𝑥 + (𝑥𝑒 𝑥 − 2𝑥)𝑑𝑦 = 0

1. (3𝑥 2 𝑦 + 𝐶𝑜𝑠 𝑦)𝑑𝑥 + (𝑥 3 − 𝑥 𝑆𝑒𝑛 𝑦) 𝑑𝑦 = 0

18. (𝑇𝑎𝑛 𝑦 + 𝑦 𝐶𝑜𝑠 𝑥)𝑑𝑥 + (𝑥𝑒 𝑥 − 2𝑥)𝑑𝑦 = 0

2. (3 𝑥𝑦 2 + 𝐶𝑜𝑠 𝑦)𝑑𝑥 + (𝑥 3 − 𝑥 𝑆𝑒𝑛 𝑦)𝑑𝑦 = 0

19. (𝑦 2 + 𝑦 𝑆𝑒𝑛 𝑥)𝑑𝑥 + (𝑆𝑒𝑛 𝑥 + 𝑥 𝑆𝑒𝑐 2 𝑦)𝑑𝑦 =0

3. (3𝑥𝑒 𝑦 + 𝑒 𝑥 𝐶𝑜𝑠 𝑦)𝑑𝑥 + (𝑥𝑒 𝑦 + 𝑒 𝑥 𝑆𝑒𝑛 𝑦)𝑑𝑦 = 0

20. (𝑦 + 𝑥𝑒 𝑥 )𝑑𝑥 + (𝑥 + 4𝑦 3 )𝑑𝑦 = 0

4. (3𝑥𝑒 𝑦 + 𝑒 𝑥 𝐶𝑜𝑠 𝑦)𝑑𝑥 + (𝑥 3 𝑒 𝑦 − 𝑒 𝑥 𝑆𝑒𝑛 𝑦)𝑑𝑦 =0 𝑥 𝑥 5. (𝑦𝑒 + 𝑥𝑦𝑒 )𝑑𝑥 + (𝐿𝑛 𝑦 + 𝑥𝑒 𝑥 + 1)𝑑𝑦 = 0 6. (𝑦𝑒 𝑥 + 𝑥𝑒 𝑥 )𝑑𝑥 + (𝐿𝑛 𝑦 + 𝑒 𝑥 + 1)𝑑𝑦 = 0 7. ( 8. (

𝑦 𝑥 2𝑦2

+1

𝑥 𝑥2𝑦2

=0

𝑦 𝑥 2𝑦2

− 2𝑥 2 ) 𝑑𝑥 + (3𝑦 +

+1

− 2𝑥) 𝑑𝑥 + (3𝑦 2 +

+1

) 𝑑𝑦

𝑥 𝑥2𝑦2

+1

) 𝑑𝑦

=0 9. (1 − 2𝑥𝑦 2 )𝑑𝑥 + (12𝑦 2 − 4𝑥𝑦)𝑑𝑦 = 0 10. (1 − 2𝑥𝑦 2 )𝑑𝑥 + (12𝑦 2 − 2𝑥 2 𝑦 + 3)𝑑𝑦 = 0

II. Resolver dadas.

las

ecuaciones

diferenciales

11. (3𝑥 2 𝑦 2 + 𝐶𝑜𝑠 𝑦 + 𝑥)𝑑𝑥 + (2𝑥 3 − 𝑥 𝑆𝑒𝑛𝑦 + 𝑦)𝑑𝑦 = 0 12. (12 𝑥 2 + 𝑦 2 + 1)𝑑𝑥 + (2𝑥𝑦 − 12𝑦 3 )𝑑𝑦 1 13. ( − 3 𝑦 2 ) 𝑑𝑧 − (6𝑥𝑦 + 𝑆𝑒𝑛 𝑦)𝑑𝑦 = 0 𝑥 14. (2𝑥 + 1)𝑑𝑥 + (5 − 6𝑦)𝑑𝑦 = 0

III. Resolver el problema de valor inicial. Utilizar un SAC para graficar la solución obtenida. 21.

(

1 1 1 + 2 ) 𝑑𝑥 + 𝑑𝑦 = 0, 2 (𝑥 + 𝑦) (𝑥 + 𝑦)2 𝑥 𝑦(1) = 0

1 𝑥+𝑦 22. ( − ) 𝑑𝑥 (𝑥 𝑥−𝑦 − 𝑦)2 𝑥+𝑦 1 +( + ) 𝑑𝑦 = 0, (𝑥 − 𝑦)2 𝑥 − 𝑦 𝑦(2) = 1 2

2

23. (2𝑒 𝑦 + 2𝑥𝑦𝑒 𝑥 𝑦 )𝑑𝑥 + (𝑥 2 𝑒 𝑥 𝑦 )𝑑𝑦 = 0, 𝑦(1) = 𝐿𝑛 𝜋 24. (𝐶𝑜𝑠 𝑦 + 𝑦 𝐶𝑜𝑡 2 𝑥)𝑑𝑥 + (𝑇𝑎𝑛 𝑥 − 𝑥 𝑆𝑒𝑛 𝑦)𝑑𝑦 𝜋 𝜋 = 0, 𝑦 ( ) = 4 4 25. (8𝑥 + 2)𝑑𝑥 + (15𝑦 2 + 3)𝑑𝑦 = 0, 𝑦(1) = 1 26. (𝑥(𝑦 + 1) + (𝑥 − 1)(𝑦 + 1))𝑑𝑥 + 𝑥(𝑥 − 1)𝑑𝑦 = 0, 𝑦(2) = 0

15. (8𝑥 − 5𝑦 + 1)𝑑𝑥 + (6𝑦 2 − 5𝑥)𝑑𝑦 = 0

ANÁLISIS MATEMÁTICA PARA INGENIEROS AGRICOLAS

LIC. DANDY ALBERT SANCHEZ ESCURRA

2 38. (𝑦 + 𝑥 𝑦 2 )𝑑𝑥 + (𝑥 2 𝑦 + 𝑥 − 𝑥𝑦)𝑑𝑦 = 0, 𝑦(𝑒) =1

27. (𝐿𝑛 𝑥 + 𝑦 𝐶𝑜𝑠 (𝑥𝑦) + 1)𝑑𝑥 + (𝑒 𝑦 + 𝑥 𝐶𝑜𝑠 (𝑥𝑦))𝑑𝑦 𝜋 = 0, 𝑦(1) = 2 28. (𝑥 3 + 𝑥 2 + 𝑦)𝑑𝑥 + 𝑥 𝑑𝑦 = 0,

39. 4 𝑑𝑥 + (2 𝑦 2 − 4𝑥 + 4𝑦)𝑑𝑦 = 0, 𝑦(2) = 1

𝑦(1) =

5 12

IV. Determinar el valor de la constante K de tal manera que la ecuación dada sea exacta y posteriormente obtener la familia de soluciones correspondientes.

40. (2𝑥 2 𝑦 3 + 𝑦)𝑑𝑥 + (2𝑥 3 𝑦 2 − 𝑥)𝑑𝑦 = 0, 𝑦(𝑒) = 𝑒

VII. Determinar una función 𝒑(𝒙, 𝒚) que haga de la ecuación dada una ecuación diferencial exacta. 41. (𝐶𝑜𝑠 𝑥 − 𝑥 𝑆𝑒𝑛 𝑥 + 3𝑦 𝐶𝑜𝑠 𝑥)𝑑𝑥 + 𝑝(𝑥, 𝑦)𝑑𝑦 =0 42. (𝑒 𝑥 −

29. (8𝑥𝑦 − 5𝑦 3 )𝑑𝑥 + (𝑘 𝑥 2 − 15𝑥𝑦 2 )𝑑𝑦 = 0 30. (8𝑦 − 72 𝑥 2 𝑦 3 + 1)𝑑𝑥 + (8𝑥 + 𝑘𝑥 3 𝑦 2 )𝑑𝑦 =0

𝑥2

𝑦 + 𝑥 𝑒 𝑥 ) 𝑑𝑥 + 𝑝(𝑥, 𝑦)𝑑𝑦 = 0 +1

43. (3 𝑥 2 𝑦 2 −

1 ) 𝑑𝑥 + 𝑝(𝑥, 𝑦)𝑑𝑦 = 0 𝑥2

31. (2 𝑆𝑒𝑛 𝑦 + 𝑘𝑒 𝑥 𝐶𝑜𝑠 𝑦)𝑑𝑥 + (2𝑥 𝐶𝑜𝑠 𝑦 + 4𝑒 𝑥 𝑆𝑒𝑛 𝑦)𝑑𝑦 = 0 32. (𝑇𝑎𝑛 𝑦 + 2𝑥𝑦)𝑑𝑥 + (𝑥 2 + 𝑘𝑥 𝑆𝑒𝑐 2 𝑦)𝑑𝑦 = 0

V. Resolver la ecuación diferencial dada determinado, si es posible, un factor integrante. 2

2

2

2

33. (𝑦𝑒 𝑦 + 2𝑥𝑒 𝑥 )𝑑𝑥 + (𝑥𝑒 𝑦 − 2𝑦𝑒 𝑥 )𝑑𝑦 = 0 2

34. 𝑥(𝑒 −𝑥 + 𝑦 2 + 1)𝑑𝑥 + 𝑦 𝑑𝑦 = 0 35. 𝑦 2 (1 + 𝐿𝑛 𝑥)𝑑𝑥 + (2 + 𝑥𝑦 𝐿𝑛 𝑥)𝑑𝑦 = 0 36. (1 + 4𝑥)𝑦 𝐿𝑛 𝑦 𝑑𝑥 + (𝑥 + 2𝑥 2 )𝑑𝑦 = 0

VI. Determinar un Factor integrante apropiado para resolver el problema de valor inicial. Graficar la solución obtenida. 37. 6𝑥𝑦 𝐿𝑛 𝑦 𝑑𝑥 + (20 𝑦 2 + 3 𝑥 2 )𝑑𝑦 = 0, 𝑦(3) = 1

ANÁLISIS MATEMÁTICA PARA INGENIEROS AGRICOLAS

LIC. DANDY ALBERT SANCHEZ ESCURRA