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# 17.1E: Ejercicios para la Sección 17.1

• Edwin “Jed” Herman & Gilbert Strang
• OpenStax

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En los ejercicios 1 - 6, clasifique cada una de las siguientes ecuaciones como lineales o no lineales. Si la ecuación es lineal, determine si es homogénea o no homogénea.

1. $$x^3y''+(x-1)y'-8y=0$$

Contestar
lineal, homogéneo

2. $$(1+y^2)y''+xy'-3y= \cos x$$

3. $$xy''+e^yy'=x$$

Contestar
no lineal

4. $$y''+ \dfrac{4}{x}y'-8xy=5x^2+1$$

5. $$y''+( \sin x)y'-xy=4y$$

Contestar
lineal, homogéneo

6. $$y''+\left(\dfrac{x+3}{y}\right)y'=0$$

En los ejercicios 7 - 10, verificar que la función dada sea una solución a la ecuación diferencial. Utilice una utilidad gráfica para graficar las soluciones particulares para varios valores de$$c_1$$ y$$c_2.$$ ¿Qué tienen en común las soluciones?

7. [T]$$y''+2y'-3y=0; \quad y(x)=c_1e^x+c_2e^{-3x}$$

8. [T]$$x^2y''-2y-3x^2+1=0; \quad y(x)=c_1x^2+c_2x^{-1}+x^2 \ln(x)+ \frac{1}{2}$$

9. [T]$$y''+14y+49y=0; \quad y(x)=c_1e^{−7x}+c_2xe^{−7x}$$

10. [T]$$6y''−49y′+8y=0; \quad y(x)=c_1e^{x/6}+c_2e^{8x}$$

En los ejercicios 11 - 30, encuentra la solución general a la ecuación diferencial lineal.

11. $$y''−3y′−10y=0$$

Contestar
$$y = c_1e^{5x} + c_2e^{-2x}$$

12. $$y''−7y′+12y=0$$

13. $$y''+4y′+4y=0$$

Contestar
$$y = c_1e^{-2x} + c_2xe^{-2x}$$

14. $$4y''−12y′+9y=0$$

15. $$2y''−3y′−5y=0$$

Contestar
$$y = c_1e^{5x/2} + c_2e^{-x}$$

16. $$3y''−14y′+8y=0$$

17. $$y''+y′+y=0$$

Contestar
$$y = e^{-x/2}\left(c_1\cos\frac{\sqrt{3}x}{2} + c_2\sin\frac{\sqrt{3}x}{2}\right)$$

18. $$5y''+2y′+4y=0$$

19. $$y''−121y=0$$

Contestar
$$y = c_1e^{-11x} + c_2e^{11x}$$

20. $$8y''+14y′−15y=0$$

21. $$y''+81y=0$$

Contestar
$$y = c_1\cos 9x + c_2\sin 9x$$

22. $$y''−y′+11y=0$$

23. $$2y''=0$$

Contestar
$$y = c_1 + c_2x$$

24. $$y''−6y′+9y=0$$

25. $$3y''−2y′−7y=0$$

Contestar
$$y = c_1e^{\left( (1+\sqrt{22})/3 \right)x} + c_2e^{\left( (1-\sqrt{22})/3 \right)x}$$

26. $$4y''−10y′=0$$

27. $$36\dfrac{d^2y}{dx^2}+12\dfrac{dy}{dx}+y=0$$

Contestar
$$y = c_1e^{-x/6} + c_2xe^{-x/6}$$

28. $$25\dfrac{d^2y}{dx^2}−80\dfrac{dy}{dx}+64y=0$$

29. $$\dfrac{d^2y}{dx^2}−9\dfrac{dy}{dx}=0$$

Contestar
$$y = c_1 + c_2e^{9x}$$

30. $$4\dfrac{d^2y}{dx^2}+8y=0$$

En los ejercicios 31 - 38, resolver el problema del valor inicial.

31. $$y''+5y′+6y=0, \quad y(0)=0,\; y′(0)=−2$$

Contestar
$$y = -2e^{-2x} + 2e^{-3x}$$

32. $$y''+2y′−8y=0, \quad y(0)=5,\; y′(0)=4$$

33. $$y''+4y=0, \quad y(0)=3, \; y′(0)=10$$

Contestar
$$y = 3\cos(2x) + 5\sin(2x)$$

34. $$y''−18y′+81y=0, \quad y(0)=1, \; y′(0)=5$$

35. $$y''−y′−30y=0, \quad y(0)=1, \; y′(0)=−16$$

Contestar
$$y = -e^{6x} + 2e^{-5x}$$

36. $$4y''+4y′−8y=0, \quad y(0)=2, \; y′(0)=1$$

37. $$25y''+10y′+y=0, \quad y(0)=2, \; y′(0)=1$$

Contestar
$$y = 2e^{-x/5} + \frac{7}{5}xe^{-x/5}$$

38. $$y''+y=0, \quad y(π)=1, \; y′(π)=−5$$

En los ejercicios 39 - 46, resolver el problema del valor límite, si es posible.

39. $$y''+y′−42y=0, \quad y(0)=0, \; y(1)=2$$

Contestar
$$y = \left( \frac{2}{e^6 - e^{-7}} \right)e^{6x} - \left( \frac{2}{e^6 - e^{-7}} \right)e^{-7x}$$

40. $$9y''+y=0, \quad y(3π^2)=6, \; y(0)=−8$$

41. $$y''+10y′+34y=0, \quad y(0)=6, \; y(π)=2$$

Contestar
No existen soluciones.

42. $$y''+7y′−60y=0, \quad y(0)=4, \; y(2)=0$$

43. $$y''−4y′+4y=0, \quad y(0)=2, \; y(1)=−1$$

Contestar
$$y = 2e^{2x} - \left( \frac{2e^{2}+1}{e^2} \right)xe^{2x}$$

44. $$y''−5y′=0, \quad y(0)=3, \; y(−1)=2$$

45. $$y''+9y=0, \quad y(0)=4, \; y(π^3)=−4$$

Contestar
$$y = 4\cos 3x + c_2\sin 3x,$$infinitamente muchas soluciones

46. $$4y''+25y=0, \quad y(0)=2, \; y(2π)=−2$$

47. Encuentre una ecuación diferencial con una solución general que sea$$y=c_1e^{x/5}+c_2e^{−4x}.$$

Contestar
$$5y'' +19y' -4y = 0$$

48. Encuentre una ecuación diferencial con una solución general que sea$$y=c_1e^{x}+c_2e^{−4x/3}.$$

Por cada ecuación diferencial en los ejercicios 49 - 51:

1. Resolver el problema de valor inicial.
2. [T] Utilice una utilidad gráfica para graficar la solución particular.

49. $$y''+64y=0; \quad y(0)=3, \; y′(0)=16$$

Contestar
a.$$y = 3\cos 8x + 2\sin 8x$$
b.

50. $$y''−2y′+10y=0; \quad y(0)=1, \; y′(0)=13$$

51. $$y''+5y′+15y=0; \quad y(0)=−2, \; y′(0)=7$$

Contestar
a.$$y = e^{-5/2}\left[-2\cos\left(\frac{\sqrt{35}}{2}x\right) + \frac{4\sqrt{35}}{35}\sin\left(\frac{\sqrt{35}}{2}x\right) \right]$$
b.

52. (Principio de superposición) Demostrar que si$$y_1(x)$$ y$$y_2(x)$$ son soluciones a una ecuación diferencial homogénea lineal,$$y''+p(x)y′+q(x)y=0,$$ entonces la función$$y(x)=c_1y_1(x)+c_2y_2(x),$$ donde$$c_1$$ y$$c_2$$ son constantes, también es una solución.

53. Demostrar que si$$a, \, b$$ y$$c$$ son constantes positivas, entonces todas las soluciones a la ecuación diferencial lineal de segundo orden se$$ay''+by′+cy=0$$ acercan a cero como$$x→∞.$$ (Pista: Considere tres casos: dos raíces distintas, raíces reales repetidas y raíces conjugadas complejas).

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