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7.7.1: The Method of Frobenius III (Exercises)

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    30760
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    Q7.7.1

    In Exercises 7.7.1-7.7.40 find a fundamental set of Frobenius solutions. Give explicit formulas for the coefficients.

    1. \(x^2y''-3xy'+(3+4x)y=0\)

    2. \(xy''+y=0\)

    3. \(4x^2(1+x)y''+4x(1+2x)y'-(1+3x)y=0\)

    4. \(xy''+xy'+y=0\)

    5. \(2x^2(2+3x)y''+x(4+21x)y'-(1-9x)y=0\)

    6. \(x^2y''+x(2+x)y'-(2-3x)y=0\)

    7. \(4x^2y''+4xy'-(9-x)y=0\)

    8. \(x^2y''+10xy'+(14+x)y=0\)

    9. \(4x^2(1+x)y''+4x(3+8x)y'-(5-49x)y=0\)

    10. \(x^2(1+x)y''-x(3+10x)y'+30xy=0\)

    11. \(x^2y''+x(1+x)y'-3(3+x)y=0\)

    12. \(x^2y''+x(1-2x)y'-(4+x)y=0\)

    13. \(x(1+x)y''-4y'-2y=0\)

    14. \(x^2(1+2x)y''+x(9+13x)y'+(7+5x)y=0\)

    15. \(4x^2y''-2x(4-x)y'-(7+5x)y=0\)

    16. \(3x^2(3+x)y''-x(15+x)y'-20y=0\)

    17. \(x^2(1+x)y''+x(1-10x)y'-(9-10x)y=0\)

    18. \(x^2(1+x)y''+3x^2y'-(6-x)y=0\)

    19. \(x^2(1+2x)y''-2x(3+14x)y'+(6+100x)y=0\)

    20. \(x^2(1+x)y''-x(6+11x)y'+(6+32x)y=0\)

    21. \(4x^2(1+x)y''+4x(1+4x)y'-(49+27x)y=0\)

    22. \(x^2(1+2x)y''-x(9+8x)y'-12xy=0\)

    23. \(x^2(1+x^2)y''-x(7-2x^2)y'+12y=0\)

    24. \(x^2y''-x(7-x^2)y'+12y=0\)

    25. \(xy''-5y'+xy=0\)

    26. \(x^2y''+x(1+2x^2)y'-(1-10x^2)y=0\)

    27. \(x^2y''-xy'-(3-x^2)y=0\)

    28. \(4x^2y''+2x(8+x^2)y'+(5+3x^2)y=0\)

    29. \(x^2y''+x(1+x^2)y'-(1-3x^2)y=0\)

    30. \(x^2y''+x(1-2x^2)y'-4(1+2x^2)y=0\)

    31. \(4x^2y''+8xy'-(35-x^2)y=0\)

    32. \(9x^2y''-3x(11+2x^2)y'+(13+10x^2)y=0\)

    33. \(x^2y''+x(1-2x^2)y'-4(1-x^2)y=0\)

    34. \(x^2y''+x(1-3x^2)y'-4(1-3x^2)y=0\)

    35. \(x^2(1+x^2)y''+x(5+11x^2)y'+24x^2y=0\)

    36. \(4x^2(1+x^2)y''+8xy'-(35-x^2)y=0\)

    37. \(x^2(1+x^2)y''-x(5-x^2)y'-(7+25x^2)y=0\)

    38. \(x^2(1+x^2)y''+x(5+2x^2)y'-21y=0\)

    39. \(x^2(1+2x^2)y''-x(3+x^2)y'-2x^2y=0\)

    40. \(4x^2(1+x^2)y''+4x(2+x^2)y'-(15+x^2)y=0\)

    Q7.7.2

    41.

    1. Under the assumptions of Theorem 7.7.1, show that \[y_1=x^{r_1}\sum_{n=0}^\infty a_n(r_1)x^n\nonumber \] and \[y_2=x^{r_2}\sum_{n=0}^{k-1}a_n(r_2)x^n+C\left(y_1\ln x+x^{r_1}\sum_{n=1}^\infty a_n'(r_1)x^n\right)\nonumber \] are linearly independent.
    2. Use the result of (a) to complete the proof of Theorem 7.7.1.

    42. Find a fundamental set of Frobenius solutions of Bessel’s equation \[x^2y''+xy'+(x^2-\nu^2)y=0\nonumber \] in the case where \(\nu\) is a positive integer.

    43. Prove Theorem 7.7.2.

    44. Under the assumptions of Theorem 7.7.1, show that \(C=0\) if and only if \(p_1(r_2+)=0\) for some integer \(\ell\) in \(\{0,1,\dots,k-1\}\).

    45. Under the assumptions of Theorem 7.7.2, show that \(C=0\) if and only if \(p_2(r_2+2)=0\) for some integer \(\ell\) in \(\{0,1,\dots,k-1\}\).

    46. Let \[Ly=\alpha_0x^2y''+\beta_0xy'+(\gamma_0+\gamma_1x)y\nonumber \] and define \[p_0(r)=\alpha_0r(r-1)+\beta_0r+\gamma_0.\nonumber \]

    Show that if \[p_0(r)=\alpha_0(r-r_1)(r-r_2)\nonumber \] where \(r_1-r_2=k\), a positive integer, then \(Ly=0\) has the solutions

    \[y_1=x^{r_1}\sum_{n=0}^\infty {(-1)^n\over n!\prod_{j=1}^n(j+k)}\left(\gamma_1\over\alpha_0\right)^n x^n\nonumber \]

    and

    \[\begin{aligned} y_2&=x^{r_2}\sum_{n=0}^{k-1} {(-1)^n\over n!\prod_{j=1}^n(j-k)} \left(\gamma_1\over\alpha_0\right)^n x^n\\[4pt]&-{1\over k!(k-1)!}\left(\gamma_1\over\alpha_0\right)^k\left(y_1\ln x- x^{r_1}\sum_{n=1}^\infty {(-1)^n\over n!\prod_{j=1}^n(j+k)}\left(\gamma_1\over\alpha_0\right)^n \left(\sum_{j=1}^n{2j+k\over j(j+k)}\right)x^n\right).\end{aligned}\nonumber \]

    47. Let \[Ly=\alpha_0x^2y''+\beta_0xy'+(\gamma_0+\gamma_2x^2)y\nonumber \] and define \[p_0(r)=\alpha_0r(r-1)+\beta_0r+\gamma_0.\nonumber \]

    Show that if \[p_0(r)=\alpha_0(r-r_1)(r-r_2)\nonumber \] where \(r_1-r_2=2k\), an even positive integer, then \(Ly=0\) has the solutions

    \[y_1=x^{r_1}\sum_{m=0}^\infty {(-1)^m\over 4^mm!\prod_{j=1}^m(j+k)}\left(\gamma_2\over\alpha_0\right)^m x^{2m}\nonumber \]

    and

    \[\begin{aligned} y_2&=x^{r_2}\sum_{m=0}^{k-1} {(-1)^m\over4^mm!\prod_{j=1}^m(j-k)} \left(\gamma_2\over\alpha_0\right)^m x^{2m}\\[4pt]&-{2\over 4^kk!(k-1)!}\left(\gamma_2\over\alpha_0\right)^k\left(y_1\ln x- {x^{r_1}\over2}\sum_{m=1}^\infty {(-1)^m\over 4^mm!\prod_{j=1}^m(j+k)}\left(\gamma_2\over\alpha_0\right)^m \left(\sum_{j=1}^m{2j+k\over j(j+k)}\right)x^{2m}\right).\end{aligned}\nonumber \]

    48. Let \(L\) be as in Exercises 7.5.57 and 7.5.58, and suppose the indicial polynomial of \(Ly=0\) is

    \[p_0(r)=\alpha_0(r-r_1)(r-r_2),\nonumber \]

    with \(k=r_1-r_2\), where \(k\) is a positive integer. Define \(a_0(r)=1\) for all \(r\). If \(r\) is a real number such that \(p_0(n+r)\) is nonzero for all positive integers \(n\), define

    \[a_n(r)=-{1\over p_0(n+r)}\sum_{j=1}^n p_j(n+r-j)a_{n-j}(r),\,n\ge1,\nonumber \]

    and let \[y_1=x^{r_1}\sum_{n=0}^\infty a_n(r_1)x^n.\nonumber \]

    Define \[a_n(r_2)=-{1\over p_0(n+r_2)}\sum_{j=1}^n p_j(n+r_2-j)a_{n-j}(r_2) \, \text{if} n\ge1 \, \text{and}\, n\ne k,\nonumber \] and let \(a_k(r_2)\) be arbitrary.

    1. Conclude from Exercise 7.6.66 that \[L\left(y_1\ln x+x^{r_1}\sum_{n=1}^\infty a_n'(r_1)x^n\right)=k\alpha_0x^{r_1}.\nonumber \]
    2. Conclude from Exercise 7.5.57 that \[L\left(x^{r_2}\sum_{n=0}^\infty a_n(r_2)x^n\right)=Ax^{r_1},\nonumber \] where \[A=\sum_{j=1}^k p_j(r_1-j)a_{k-j}(r_2).\nonumber \]
    3. Show that \(y_1\) and \[y_2=x^{r_2}\sum_{n=0}^\infty a_n(r_2)x^n -{A\over k\alpha_0} \left(y_1 \ln x+x^{r_1}\sum_{n=1}^\infty a_n'(r_1)x^n\right)\nonumber \] form a fundamental set of Frobenius solutions of \(Ly=0\).
    4. Show that choosing the arbitrary quantity \(a_k(r_2)\) to be nonzero merely adds a multiple of \(y_1\) to \(y_2\). Conclude that we may as well take \(a_k(r_2)~=~0\).

    This page titled 7.7.1: The Method of Frobenius III (Exercises) is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by William F. Trench.

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