7.4.1: Series Solutions Near an Ordinary Point II (Exercises)

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$trench-1995.jpg$
William F. Trench
Andrew G. Cowles Distinguished Professor Emeritus (Mathematics) at Trinity University

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Q7.3.1

In Exercises 7.3.1-7.3.12 find the coefficients $a_0$,…, $a_N$ for $N$ at least $7$ in the series solution $y=\sum_{n=0}^\infty a_nx^n$ of the initial value problem.

1. $(1+3x)y''+xy'+2y=0,\quad y(0)=2,\quad y'(0)=-3$

2. $(1+x+2x^2)y''+(2+8x)y'+4y=0,\quad y(0)=-1,\quad y'(0)=2$

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

4. $(1+x+3x^2)y''+(2+15x)y'+12y=0,\quad y(0)=0,\quad y'(0)=1$

5. $(2+x)y''+(1+x)y'+3y=0,\quad y(0)=4,\quad y'(0)=3$

6. $(3+3x+x^2)y''+(6+4x)y'+2y=0,\quad y(0)=7,\quad y'(0)=3$

7. $(4+x)y''+(2+x)y'+2y=0,\quad y(0)=2,\quad y'(0)=5$

8. $(2-3x+2x^2)y''-(4-6x)y'+2y=0,\quad y(1)=1,\quad y'(1)=-1$

9. $(3x+2x^2)y''+10(1+x)y'+8y=0,\quad y(-1)=1,\quad y'(-1)=-1$

10. $(1-x+x^2)y''-(1-4x)y'+2y=0,\quad y(1)=2,\quad y'(1)=-1$

11. $(2+x)y''+(2+x)y'+y=0,\quad y(-1)=-2,\quad y'(-1)=3$

12. $x^2y''-(6-7x)y'+8y=0,\quad y(1)=1,\quad y'(1)=-2$

Q7.3.2

13. Do the following experiment for various choices of real numbers $a_0$, $a_1$, and $r$, with $0<r<1/\sqrt2$.

Use differential equations software to solve the initial value problem \[(1+x+2x^2)y''+(1+7x)y'+2y=0,\quad y(0)=a_0,\quad y'(0)=a_1, \tag{A}\] numerically on $(-r,r)$. (See Example 7.3.1.)
For $N=2$, $3$, $4$, …, compute $a_2$, …, $a_N$ in the power series solution $y=\sum_{n=0}^\infty a_nx^n$ of (A), and graph \[T_N(x)=\sum_{n=0}^N a_nx^n\nonumber \] and the solution obtained in (a) on $(-r,r)$. Continue increasing $N$ until there’s no perceptible difference between the two graphs.

14. Do the following experiment for various choices of real numbers $a_0$, $a_1$, and $r$, with $0<r<2$.

Use differential equations software to solve the initial value problem \[(3+x)y''+(1+2x)y'-(2-x)y=0,\quad y(-1)=a_0,\quad y'(-1)=a_1, \tag{A}\] numerically on $(-1-r,-1+r)$. (See Example 7.3.2). Why this interval?)
For $N=2$, $3$, $4$, …, compute $a_2,\dots,a_N$ in the power series solution \[y=\sum_{n=0}^\infty a_n(x+1)^n\nonumber \] of (A), and graph \[T_N(x)=\sum_{n=0}^N a_n(x+1)^n\nonumber \] and the solution obtained in (a) on $(-1-r,-1+r)$. Continue increasing $N$ until there’s no perceptible difference between the two graphs.

15. Do the following experiment for several choices of $a_0$, $a_1$, and $r$, with $r>0$.

Use differential equations software to solve the initial value problem \[y''+3xy'+(4+2x^2)y=0,\quad y(0)=a_0,\quad y'(0)=a_1, \tag{A}\] numerically on $(-r,r)$. (See Example 7.3.3.)
Find the coefficients $a_0$, $a_1$, …, $a_N$ in the power series solution $y=\sum_{n=0}^\infty a_nx^n$ of (A), and graph \[T_N(x)=\sum_{n=0}^N a_nx^n\nonumber \] and the solution obtained in (a) on $(-r,r)$. Continue increasing $N$ until there’s no perceptible difference between the two graphs.

16. Do the following experiment for several choices of $a_0$ and $a_1$.

Use differential equations software to solve the initial value problem \[(1-x)y''-(2-x)y'+y=0,\quad y(0)=a_0,\quad y'(0)=a_1, \tag{A}\] numerically on $(-r,r)$.
Find the coefficients $a_0$, $a_1$, …, $a_N$ in the power series solution $y=\sum_{n=0}^Na_nx^n$ of (A), and graph \[T_N(x)=\sum_{n=0}^N a_nx^n\nonumber \] and the solution obtained in (a) on $(-r,r)$. Continue increasing $N$ until there’s no perceptible difference between the two graphs. What happens as you let $r\to 1$?

17. Follow the directions of Exercise 7.3.16 for the initial value problem \[(1+x)y''+3y'+32y=0,\quad y(0)=a_0,\quad y'(0)=a_1.\nonumber \]

18. Follow the directions of Exercise 7.3.16 for the initial value problem \[(1+x^2)y''+y'+2y=0,\quad y(0)=a_0,\quad y'(0)=a_1.\nonumber \]

Q7.3.3

In Exercises 7.3.19-7.3.28 find the coefficients $a_{0},...a_{N}$ for $N$ at least $7$ in the series solution \[y=\sum_{n=0}^{\infty}a_{n}(x-x_{0})^{n}\nonumber \] of the initial value problem. Take $x_{0}$ to be the point where the initial conditions are imposed.

19. $(2+4x)y''-4y'-(6+4x)y=0,\quad y(0)=2,\quad y'(0)=-7$

20. $(1+2x)y''-(1-2x)y'-(3-2x)y=0,\quad y(1)=1,\quad y'(1)=-2$

21. $(5+2x)y''-y'+(5+x)y=0,\quad y(-2)=2,\quad y'(-2)=-1$

22. $(4+x)y''-(4+2x)y'+(6+x)y=0,\quad y(-3)=2,\quad y'(-3)=-2$

23. $(2+3x)y''-xy'+2xy=0,\quad y(0)=-1,\quad y'(0)=2$

24. $(3+2x)y''+3y'-xy=0,\quad y(-1)=2,\quad y'(-1)=-3$

25. $(3+2x)y''-3y'-(2+x)y=0,\quad y(-2)=-2,\quad y'(-2)=3$

26. $(10-2x)y''+(1+x)y=0,\quad y(2)=2,\quad y'(2)=-4$

27. $(7+x)y''+(8+2x)y'+(5+x)y=0,\quad y(-4)=1,\quad y'(-4)=2$

28. $(6+4x)y''+(1+2x)y=0,\quad y(-1)=-1,\quad y'(-1)=2$

Q7.3.4

29. Show that the coefficients in the power series in $x$ for the general solution of \[(1+\alpha x+\beta x^2)y''+(\gamma+\delta x)y'+\epsilon y=0\nonumber \] satisfy the recurrrence relation \[a_{n+2}=-{\gamma+\alpha n\over n+2}\,a_{n+1}-{\beta n(n-1)+\delta n+\epsilon\over(n+2)(n+1)}\, a_n.\nonumber \]

30.

Let $\alpha$ and $\beta$ be constants, with $\beta\ne0$. Show that $y=\sum_{n=0}^\infty a_nx^n$ is a solution of \[(1+\alpha x+\beta x^2)y''+(2\alpha+4\beta x)y'+2\beta y=0 \tag{A}\] if and only if \[a_{n+2}+\alpha a_{n+1}+\beta a_n=0,\quad n\ge0. \tag{B}\] An equation of this form is called a second order homogeneous linear difference equation. The polynomial $p(r)=r^2+\alpha r+\beta$ is called the characteristic polynomial of (B). If $r_1$ and $r_2$ are the zeros of $p$, then $1/r_1$ and $1/r_2$ are the zeros of \[P_{0}(x)=1+\alpha x+\beta x^{2}\nonumber \]
Suppose $p(r)=(r-r_1)(r-r_2)$ where $r_1$ and $r_2$ are real and distinct, and let $\rho$ be the smaller of the two numbers $\{1/|r_1|,1/|r_2|\}$. Show that if $c_1$ and $c_2$ are constants then the sequence \[a_n=c_1r_1^n+c_2r_2^n,\quad n\ge0\nonumber \] satisfies (B). Conclude from this that any function of the form \[y=\sum_{n=0}^\infty (c_1r_1^n+c_2r_2^n)x^n\nonumber \] is a solution of (A) on $(-\rho,\rho)$.
Use (b) and the formula for the sum of a geometric series to show that the functions \[y_1={1\over1-r_1x}\quad\mbox{ and }\quad y_2={1\over1-r_2x}\nonumber \] form a fundamental set of solutions of (A) on $(-\rho,\rho)$.
Show that $\{y_1,y_2\}$ is a fundamental set of solutions of (A) on any interval that does’nt contain either $1/r_1$ or $1/r_2$.
Suppose $p(r)=(r-r_1)^2$, and let $\rho=1/|r_1|$. Show that if $c_1$ and $c_2$ are constants then the sequence \[a_n=(c_1+c_2n)r_1^n,\quad n\ge0\nonumber \] satisfies (B). Conclude from this that any function of the form \[y=\sum_{n=0}^\infty (c_1+c_2n)r_1^nx^n\nonumber \] is a solution of (A) on $(-\rho,\rho)$.
Use (e) and the formula for the sum of a geometric series to show that the functions \[y_1={1\over1-r_1x}\quad\mbox{ and }\quad y_2={x\over(1-r_1x)^2}\nonumber \] form a fundamental set of solutions of (A) on $(-\rho,\rho)$.
Show that $\{y_1,y_2\}$ is a fundamental set of solutions of (A) on any interval that does not contain $1/r_1$.

31. Use the results of Exercise 7.3.30 to find the general solution of the given equation on any interval on which polynomial multiplying $y''$ has no zeros.

$(1+3x+2x^2)y''+(6+8x)y'+4y=0$
$(1-5x+6x^2)y''-(10-24x)y'+12y=0$
$(1-4x+4x^2)y''-(8-16x)y'+8y=0$
$(4+4x+x^2)y''+(8+4x)y'+2y=0$
$(4+8x+3x^2)y''+(16+12x)y'+6y=0$

Q7.3.5

In Exercises 7.3.32-7.3.38 find the coefficients $a_{0}, ..., a_{N}$ for $N$ at least $7$ in the series solution $y=\sum_{n=0}^{\infty} a_{n}x^{n}$ of the initial value problem.

32. $y''+2xy'+(3+2x^2)y=0,\quad y(0)=1,\quad y'(0)=-2$

33. $y''-3xy'+(5+2x^2)y=0,\quad y(0)=1,\quad y'(0)=-2$

34. $y''+5xy'-(3-x^2)y=0,\quad y(0)=6,\quad y'(0)=-2$

35. $y''-2xy'-(2+3x^2)y=0,\quad y(0)=2,\quad y'(0)=-5$

36. $y''-3xy'+(2+4x^2)y=0,\quad y(0)=3,\quad y'(0)=6$

37. $2y''+5xy'+(4+2x^2)y=0,\quad y(0)=3,\quad y'(0)=-2$

38. $3y''+2xy'+(4-x^2)y=0,\quad y(0)=-2,\quad y'(0)=3$

Q7.3.6

39. Find power series in $x$ for the solutions $y_1$ and $y_2$ of \[y''+4xy'+(2+4x^2)y=0\nonumber \] such that $y_1(0)=1$, $y'_1(0)=0$, $y_2(0)=0$, $y'_2(0)=1$, and identify $y_1$ and $y_2$ in terms of familiar elementary functions.

Q7.3.7

In Exercises 7.3.40-7.3.49 find the coefficients $a_{0}, ..., a_{N}$ for $N$ at least $7$ in the series solution \[y=\sum_{n=0}^{\infty} a_{n}(x-x_{0})^{n}\nonumber \] of the initial value problem. Take $x_{0}$ to be the point where the initial conditions are imposed.

40. $(1+x)y''+x^2y'+(1+2x)y=0,\quad y(0)-2,\quad y'(0)=3$

41. $y''+(1+2x+x^2)y'+2y=0,\quad y(0)=2,\quad y'(0)=3$

42. $(1+x^2)y''+(2+x^2)y'+xy=0,\quad y(0)=-3,\quad y'(0)=5$

43. $(1+x)y''+(1-3x+2x^2)y'-(x-4)y=0,\quad y(1)=-2,\quad y'(1)=3$

44. $y''+(13+12x+3x^2)y'+(5+2x),\quad y(-2)=2,\quad y'(-2)=-3$

45. $(1+2x+3x^2)y''+(2-x^2)y'+(1+x)y=0,\quad y(0)=1,\quad y'(0)=-2$

46. $(3+4x+x^2)y''-(5+4x-x^2)y'-(2+x)y=0,\quad y(-2)=2,\quad y'(-2)=-1$

47. $(1+2x+x^2)y''+(1-x)y=0,\quad y(0)=2,\quad y'(0)=-1$

48. $(x-2x^2)y''+(1+3x-x^2)y'+(2+x)y=0,\quad y(1)=1,\quad y'(1)=0$

49. $(16-11x+2x^2)y''+(10-6x+x^2)y'-(2-x)y,\quad y(3)=1,\quad y'(3)=-2$