25: Section 7.2 Answers
- Page ID
- 103659
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)1. \(y=e^{x}(c_{1}+c_{2}x+c_{3}x^{2})\)
2. \(y=c_{1}e^{x}+c_{2}e^{-x}+c_{3}\cos 3x+c_{4}\sin 3x\)
3. \(y = c_{1}e^{x} + c_{2} \cos 4x + c_{3} \sin 4x\)
4. \(y = c_{1}e^{x} + c_{2}e^{−x} + c_{3}e^{−3x/2}\)
5. \(y = c_{1}e^{−x} + e^{−2x} (c_{1} \cos x + c_{2} \sin x)\)
6. \(y = c_{1}e^{x} + e^{x/2} (c_{2} + c_{3}x)\)
7. \(y = e ^{-x/3} (c_{1} + c_{2}x + c_{3}x^{2} )\)
8. \(y = c_{1} + c_{2}x + c_{3} \cos x + c_{4} \sin x\)
9. \(y = c_{1}e^{2x} + c_{2}e^{−2x} + c_{3} \cos 2x + c_{4} \sin 2x\)
10. \(y = (c_{1} + c_{2}x) \cos \sqrt{6}x + (c_{3} + c_{4}x) \sin\sqrt{6}x\)
11. \(y = e^{3x/2} (c_{1} + c_{2}x) + e^{−3x/2} (c_{3} + c_{4}x)\)
12. \(y = c_{1}e^{−x/2} + c_{2}e^{−x/3} + c_{3} \cos x + c_{4} \sin x\)
13. \(y = c_{1}e^{x}+c_{2}e^{−2x}+c_{3}e^{−x/2}+c_{4}e^{−3x/2}\)
14. \(y = e^{x} (c_{1}+c_{2}x+c_{3} \cos x+c_{4} \sin x)\)
15. \(y = \cos 2x − 2 \sin 2x + e^{2x}\)
16. \(y = 2e^{x} + 3e^{−x} − 5e^{−3x}\)
17. \(y = 2e^{x} + 3xe^{x} − 4e^{−x}\)
18. \(y = 2e^{−x} \cos x − 3e^{−x} \sin x + 4e^{2x}\)
19. \(y = \frac{9}{5} e^{−5x/3} + e^{x} (1 + 2x)\)
20. \(y = e^{2x} (1 − 3x + 2x^{2} )\)
21. \(y = e^{3x} (2 − x) + 4e^{−x/2}\)
22. \(y = e^{x/2} (1 − 2x) + 3e^{−x/2}\)
23. \(y = \frac{1}{8} (5e^{2x} + e^{−2x} + 10 \cos 2x + 4 \sin 2x)\)
24. \(y = −4e^{x} + e^{2x} − e^{4x} + 2e^{−x}\)
25. \(y=2e^{x}-e^{-x}\)
26. \(y = e^{2x} + e^{−2x} + e^{−x} (3 \cos x + \sin x)\)
27. \(y = 2e^{−x/2} + \cos 2x − \sin 2x\)
28. \(\{e^{x},xe^{x},e^{2x}\}\:\:1\)
29. \(\{\cos 2x, \sin 2x, e^{3x} \}\: :\: 26\)
30. \(\{e ^{−x} \cos x, e^{−x} \sin x, e^{x} \}\: :\: 5\)
31. \(\{1, x, x^{2}, e^{x} \}\: 2e^{x}\)
32. \(\{e^{x}, e^{−x}, \cos x, \sin x \}\:8\)
33. \(\{\cos x, \sin x, e^{x} \cos x, e^{x} \sin x\}\: :\: 5\)
34. \(\{e^{−3x} \cos 2x, e^{−3x} \sin 2x, e^{2x}, xe^{2x}, 1, x, x^{2} \}\)
35. \(\{e^{x}, xe^{x}, e^{x/2}, xe^{x/2}, x^{2} e^{x/2}, \cos x, \sin x \}\)
36. \(\{\cos 3x, x \cos 3x, x^{2} \cos 3x, \sin 3x, x \sin 3x, x^{2} \sin 3x, 1, x \}\)
37. \(\{e^{2x}, xe^{2x}, x^{2} e^{2x}, e^{−x}, xe^{−x}, 1 \}\)
38. \(\{\cos x, \sin x, \cos 3x, x \cos 3x, \sin 3x, x \sin 3x, e^{2x} \}\)
39. \(\{e^{2x}, xe^{2x}, e^{−2x}, xe^{−2x}, \cos 2x, x \cos 2x, \sin 2x, x \sin 2x \}\)
40. \(\{e^{−x/2} \cos \sqrt{2}x, xe^{−x/2} \cos \sqrt{2}x, x^{2} e^{−x/2} \cos \sqrt{2}x, e^{−x/2} \sin \sqrt{2}x, xe^{−x/2} \sin \sqrt{2}x, x^{2} e^{−x/2} \sin \sqrt{2}x \}\)
41. \(\{1, x, x^{2}, e^{2x}, xe^{2x}, \cos 2x, x \cos 2x, \sin 2x, x \sin 2x \}\)
42. \(\{\cos (x/2), x \cos (x/2), \sin (x/2), x \sin (x/2), \cos 2x/3 x \cos (2x/3), x^{2} \cos (2x/3), \sin (2x/3), x \sin (2x/3), x^{2} \sin (2x/3) \}\)
43. \(y=c_1x^{-2}+c_2\cos (2\ln x)+c_3\sin (2\ln x)\)
44. \(y=c_1x^3+c_2\cos (\sqrt{2}\ln x)+c_3\sin (\sqrt{2}\ln x)\)
45. \(y=c_1x+c_2x\ln x+c_3x\ln^2 x\)
46. \(y=c_1+c_2x+c_3x^2+c_4x^{-3}\)
47. \(y=c_1\cos (\ln x)+c_2\sin (\ln x)+c_3\ln x \cos (\ln x)+c_4\ln x \sin (\ln x)\)
48. \(y=c_1x+c_2x^2+c_3x^3\)