17.1E: Mazoezi ya Sehemu ya 17.1

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Edwin “Jed” Herman & Gilbert Strang
OpenStax

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Katika mazoezi ya 1 - 6, weka kila moja ya equations zifuatazo kama linear au nonlinear. Ikiwa equation ni mstari, onyesha ikiwa ni sawa au isiyo ya kawaida.

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

Jibu: linear, homogenous

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

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

Jibu: isiyo ya moja kwa moja

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

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

Jibu: linear, homogenous

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

Katika mazoezi ya 7 - 10, hakikisha kwamba kazi iliyotolewa ni suluhisho la usawa tofauti. Tumia matumizi ya graphing kwa graph ufumbuzi maalum kwa maadili kadhaa ya$c_1$ na$c_2.$ Je, ufumbuzi una sawa?

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]$6y''−49y′+8y=0; \quad y(x)=c_1e^{x/6}+c_2e^{8x}$

Katika mazoezi 11 - 30, pata suluhisho la jumla kwa equation tofauti ya mstari.

11. $y''−3y′−10y=0$

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

12. $y''−7y′+12y=0$

13. $y''+4y′+4y=0$

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

14. $4y''−12y′+9y=0$

15. $2y''−3y′−5y=0$

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

16. $3y''−14y′+8y=0$

17. $y''+y′+y=0$

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

18. $5y''+2y′+4y=0$

19. $y''−121y=0$

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

20. $8y''+14y′−15y=0$

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

Jibu: $y = c_1\cos 9x + c_2\sin 9x$

22. $y''−y′+11y=0$

23. $2y''=0$

Jibu: $y = c_1 + c_2x$

24. $y''−6y′+9y=0$

25. $3y''−2y′−7y=0$

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

26. $4y''−10y′=0$

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

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

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

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

Jibu: $y = c_1 + c_2e^{9x}$

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

Katika mazoezi 31 - 38, tatua tatizo la thamani ya awali.

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

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

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

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

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

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

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

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

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

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

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

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

Katika mazoezi 39 - 46, tatua tatizo la thamani ya mipaka, ikiwa inawezekana.

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

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

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

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

Jibu: Hakuna ufumbuzi uliopo.

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

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

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

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

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

Jibu: $y = 4\cos 3x + c_2\sin 3x,$ufumbuzi mkubwa sana

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

47. Pata equation tofauti na ufumbuzi wa jumla yaani$y=c_1e^{x/5}+c_2e^{−4x}.$

Jibu: $5y'' +19y' -4y = 0$

48. Pata equation tofauti na ufumbuzi wa jumla yaani$y=c_1e^{x}+c_2e^{−4x/3}.$

Kwa kila equation tofauti katika mazoezi 49 - 51:

Tatua tatizo la thamani ya awali.
[T] Tumia matumizi ya graphing kwa graph suluhisho fulani.

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

Jibu: a.$y = 3\cos 8x + 2\sin 8x$
b.
$Takwimu hii ni grafu ya mara kwa mara. Ina amplitude ya 3.5. Wote x na y axes ni kuongezwa katika nyongeza ya 1.$

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

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

Jibu: 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.
$Takwimu hii ni grafu ya kazi ya oscillating. Axes x na y zinaongezwa kwa nyongeza za idadi hata. Amplitude ya grafu inapungua kama ongezeko la x.$

52. (Kanuni ya superposition) Thibitisha kwamba kama$y_1(x)$ na$y_2(x)$ ni ufumbuzi wa equation linear homogeneous tofauti,$y''+p(x)y′+q(x)y=0,$ basi kazi$y(x)=c_1y_1(x)+c_2y_2(x),$ ambapo$c_1$ na$c_2$ ni mara kwa mara, pia ni suluhisho.

53. Kuthibitisha kwamba kama$a, \, b$ na$c$ ni chanya constants, basi ufumbuzi wote wa pili ili linear tofauti equation$ay''+by′+cy=0$ mbinu sifuri kama$x→∞.$ (Kidokezo: Fikiria kesi tatu: mbili mizizi tofauti, mara kwa mara mizizi halisi, na tata conjugate mizizi.)