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# 17.2E: Ejercicios para la Sección 17.2

• Edwin “Jed” Herman & Gilbert Strang
• OpenStax

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Resuelve las siguientes ecuaciones utilizando el método de coeficientes indeterminados.

1. $$2y''−5y′−12y=6$$

2. $$3y''+y′−4y=8$$

Responder
$$y=c_1e^{−4x/3}+c_2e^x−2$$

3. $$y''−6y′+5y=e^{−x}$$

4. $$y''+16y=e^{−2x}$$

Responder
$$y=c_1 \cos4x+c_2 \sin 4x+\frac{1}{20}e^{−2x}$$

5. $$y″−4y=x^2+1$$

6. $$y″−4y′+4y=8x^2+4x$$

Responder
$$y=c_1e^{2x}+c_2xe^{2x}+2x^2+5x$$

7. $$y″−2y′−3y= \sin 2x$$

8. $$y″+2y′+y= \sin x+ \cos x$$

Responder
$$y=c_1e^{−x}+c_2xe^{−x}+\frac{1}{2} \sin x−\frac{1}{2} \cos x$$

9. $$y″+9y=e^x \cos x$$

10. $$y″+y=3 \sin 2x+x \cos 2x$$

Responder
$$y=c_1 \cos x+ c_2 \sin x−\frac{1}{3}x \cos 2x−\frac{5}{9} \sin 2x$$

11. $$y″+3y′−28y=10e^{4x}$$

12. $$y″+10y′+25y=xe^{−5x}+4$$

Responder
$$y=c_1e^{−5x}+c_2xe^{−5x}+\frac{1}{6}x^3e^{−5x}+\frac{4}{25}$$

En los ejercicios 13 a 18,

1. Escribir el formulario para la solución particular$$y_p(x)$$ para el método de coeficientes indeterminados.
2. [T] Utilice un sistema de álgebra computacional para encontrar una solución particular a la ecuación dada.

13. $$y″−y′−y=x+e^{−x}$$

14. $$y″−3y=x^2−4x+11$$

Responder

a.$$y_p(x)=Ax^2+Bx+C$$

b.$$y_p(x)=−\frac{1}{3}x^2+\frac{4}{3}x−\frac{35}{9}$$

15. $$y''−y′−4y=e^x \cos 3x$$

16. $$2y″−y′+y=(x^2−5x)e^{−x}$$

Responder

a.$$y_p(x)=(Ax^2+Bx+C)e^{−x}$$

b.$$y_p(x)=(\frac{1}{4}x^2−\frac{5}{8}x−\frac{33}{32})e^{−x}$$

17. $$4y″+5y′−2y=e^{2x}+x \sin x$$

18. $$y''−y′−2y=x^2e^x \sin x$$

Responder

a.$$y_p(x)=(Ax^2+Bx+C)e^x \cos x+(Dx^2+Ex+F)e^x \sin x$$

b.$$y_p(x)=(−\frac{1}{10}x^2−\frac{11}{25}x−\frac{27}{250})e^x \cos x +(−\frac{3}{10}x^2+\frac{2}{25}x+\frac{39}{250})e^x \sin x$$

Resolver la ecuación diferencial utilizando ya sea el método de coeficientes indeterminados o la variación de parámetros.

19. $$y″+3y′−4y=2e^x$$

20. $$y''+2y′=e^{3x}$$

Responder
$$y=c_1+c_2e^{−2x}+\frac{1}{15}e^{3x}$$

21. $$y''+6y′+9y=e^{−x}$$

22. $$y''+2y′−8y=6e^{2x}$$

Responder
$$y=c_1e^{2x}+c_2e^{−4x}+xe^{2x}$$

Resolver la ecuación diferencial utilizando el método de variación de parámetros.

23. $$4y″+y=2 \sin x$$

24. $$y″−9y=8x$$

Responder
$$y=c_1e^{3x}+c_2e^{−3x}−\frac{8x}{9}$$

25. $$y″+y= \sec x, \quad 0<x<π/2$$

26. $$y″+4y=3 \csc 2x, \quad 0<x<π/2$$

Responder
$$y=c_1 \cos 2x+c_2 \sin 2x−\frac{3}{2} x \cos 2x+\frac{3}{4} \sin 2x \ln ( \sin 2x)$$

Encuentre la solución única que satisfaga la ecuación diferencial y las condiciones iniciales dadas, donde$$y_p(x)$$ está la solución particular.

27. $$y″−2y′+y=12e^x,\quad y_p(x)=6x^2e^x, \; y(0)=6, \; y′(0)=0$$

28. $$y''−7y′=4xe^{7x},\quad y_p(x)=\frac{2}{7}x^2e^{7x}−\frac{4}{49}xe^{7x}, \; y(0)=−1, \; y'(0)=0$$

Responder
$$y=− \frac {347}{343}+ \frac {4}{343}e^{7x}+\frac{2}{7}x^2e^{7x}−\frac{4}{49}xe^{7x}$$

29. $$y″+y= \cos x−4 \sin x, \quad y_p(x)=2x \cos x+\frac{1}{2} x \sin x, \; y(0)=8, \; y′(0)=−4$$

30. $$y″−5y′=e^{5x}+8e^{−5x}, \quad y_p(x)=\frac{1}{5}xe^{5x}+\frac{4}{25}e^{−5x}, \; y(0)=−2, \; y′(0)=0$$

Responder
$$y=−\frac{57}{25}+\frac{3}{25}e^{5x}+\frac{1}{5}xe^{5x}+\frac{4}{25}e^{−5x}$$

En los problemas 31 - 32 se dan dos soluciones linealmente independientes$$y_1$$$$y_2$$ —y —que satisfacen la ecuación homogénea correspondiente. Utilice el método de variación de parámetros para encontrar una solución particular a la ecuación no homogénea dada. Asumir$$x>0$$ en cada ejercicio.

31. $$x^2y″+2xy′−2y=3x, \quad y_1(x)=x, \; y2(x)=x^{−2}$$

32. $$x^2y''−2y=10x^2−1,\quad y_1(x)=x^2, \; y_2(x)=x^{−1}$$

Responder
$$y_p=\frac{1}{2}+\frac{10}{3}x^2 \ln x$$

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