5.7.1: Cauchy-Euler Equations (Exercises)
- Page ID
- 108311
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In Exercises 1-40 find the general solution of the given equation on \((0,\infty)\).
1. \(x^2y''+7xy'+8y=0\)
2. \(x^2y''-7xy'+7y=0\)
3. \(x^2y''-xy'+y=0\)
4. \(x^2y''+5xy'+4y=0\)
5. \(x^2y''+xy'+y=0\)
6. \(x^2y''-3xy'+13y=0\)
7. \(x^2y''+3xy'-3y=0\)
8. \(12x^2y''-5xy''+6y=0\)
9. \(4x^2y''+8xy'+y=0\)
10. \(3x^2y''-xy'+y=0\)
11. \(2x^2y''-3xy'+2y=0\)
12. \(x^2y''+3xy'+5y=0\)
13. \(9x^2y''+15xy'+y=0\)
14. \(x^2y''-xy'+10y=0\)
15. \(x^2y''-6y=0\)
16. \(2x^2y''+3xy'-y=0\)
17. \(x^2y''-3xy'+4y=0\)
18. \(2x^2y''+10xy'+9y=0\)
19. \(x^2y''-2y=0\)
20. \(xy''+y'=0\)
21. \(x^2y''+xy'+4y=0\)
22. \(x^{2} y^{\prime \prime}-3 x y^{\prime}-2 y=0\)
23. \(25x^2y''+25xy'+y=0\)
24. \(x^{2} y^{\prime \prime}+5 x y^{\prime}+4 y=0\)
25. \(3x^2y''+6xy'+y=0\)
26. \(x^{2} y^{\prime \prime}+3 x y^{\prime}-3 y=0\)
27. \(2 x^{2} y^{\prime \prime}+5 x y^{\prime}+y=0\)
28. \(4 x^{2} y^{\prime \prime}+y=0\)
29. \(xy''-4y'=x^4\)
30. \(x^2y''-xy'+y=2x\)
31. \(x^{2} y^{\prime \prime}- x y^{\prime}-3 y=2 x^{2}\)
32. \(x^{2} y^{\prime \prime}- x y^{\prime}-3 y=2 x^{3}\)
33. \(x^2y''-4xy'+6y=2x^4+x^2\)
34. \(x^2y''-3xy'+3y=2x^4e^x\)
35. \(x^2y''-xy'+y=\ln x\)
36. \(2 x^{2} y^{\prime \prime}+5 x y^{\prime}+y=x^{2}+x\)
37. \(x^{2} y^{\prime \prime}+5 x y^{\prime}+4 y=2 x^{3}\)
38. \(x^2y''+10xy'+8y=x^2\)
39. \(x^2y''-4xy'+6y=\ln x^2\)
40. \(x^2y''-3xy'+13y=4+3x\)
In Exercises 41-49 find the solution of the given initial value problem on \((0,\infty)\).
41. \(x^2y''+3xy'=0; \quad y(1)=0, \quad y'(1)=4\)
42. \(x^2y''-5xy'+8y=0; \quad y(2)=32, \quad y'(2)=0\)
43. \(x^2y''+xy'+y=0; \quad y(1)=1, \quad y'(1)=2\)
44. \(x^2y''-3xy'+4y=0; \quad y(1)=5, \quad y'(1)=3\)
45. \(x^2y''+3xy'+y=0; \quad y(1)=0, \quad y'(1)=1\)
46. \(xy''+y'=x; \quad y(1)=1, \quad y'(1)=-1/2\)
47. \(x^2y''-5xy'+8y=8x^6; \quad y({1\over 2})=0, \quad y'({1\over 2})=0\)
48. \(x^2y''-3xy'+3y=2x^4e^x; \quad y(1)=0, \quad y'(1)=0\)
49. \(x^2y''-xy'+y=\ln x; \quad y(1)=2, \quad y'(1)=2\)
50. Prove that if \(y=y(x)\) satisfies the Cauchy-Euler equation \[ax^2{d^2y\over dx^2}+bx{dy\over dx}+cy=0\tag{A}\] on \((0,\infty)\) then \(y=y(t)\), where \(t=-x\), satisfies A on \((-\infty,0)\).
In Exercises 51-52 find the solution of the given initial value problem on \((-\infty,0)\).
51. \(4x^2y''+y=0; \quad y(-1)=2, \quad y'(-1)=4\)
52. \(x^2y''-4xy'+6y=0; \quad y(-2)=8, \quad y'(-2)=0\)
53. A nontrivial solution of
\[P_0(x)y''+P_1(x)y'+P_2(x)y=0\nonumber\]
is said to be oscillatory on an interval \((a,b)\) if it has infinitely many zeros on \((a,b)\). Otherwise \(y\) is said to be nonoscillatory on \((a,b)\). Prove that the equation
\[x^2y''+ky=0 \quad (k=\; \mbox{constant})\nonumber\]
has oscillatory solutions on \((0,\infty)\) if and only if \(k>1/4\).