6.3.1: Series Solutions Near an Ordinary Point I (Exercises)
- Page ID
- 43318
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Q6.3.1
In Exercises 6.3.1-6.3.8 find the general solution about the ordinary point x = 0. Also, find a fundamental set of solutions.
1. \((1+x^2)y''+6xy'+6y=0\)
2. \((1+x^2)y''+2xy'-2y=0\)
3. \((1+x^2)y''-8xy'+20y=0\)
4. \((1-x^2)y''-8xy'-12y=0\)
5. \((1+2x^2)y''+7xy'+2y=0\)
6. \({(1+x^2)y''+2xy'+{1\over4}y=0}\)
7. \((x+2)y''+xy'-y=0\)
8. \((1+x^2)y''-10xy'+28y=0\)
Q6.3.2
In Exercises 6.3.11-6.3.13 find \(a_{0}, ..., a_{N}\) for \(N\) at least \(7\) in the power series solution \(y=\sum _{n=0}^{\infty} a_{n}x^{n}\) of the initial value problem.
9. \((1+x^2)y''+xy'+y=0,\quad y(0)=2,\quad y'(0)=-1\)
10. \((1+2x^2)y''-9xy'-6y=0,\quad y(0)=1,\quad y'(0)=-1\)
11. \((1+8x^2)y''+2y=0,\quad y(0)=2,\quad y'(0)=-1\)
Q6.3.3
In Exercises 6.3.12-6.3.16 find the power series in \(x-x_{0}\) for the general solution.
12. \(y''-y=0;\quad x_0=3\)
13. \(y''-(x-3)y'-y=0;\quad x_0=3\)
14. \((1-4x+2x^2)y''+10(x-1)y'+6y=0;\quad x_0=1\)
15. \((11-8x+2x^2)y''-16(x-2)y'+36y=0;\quad x_0=2\)
16. \((5+6x+3x^2)y''+9(x+1)y'+3y=0;\quad x_0=-1\)
Q6.3.4
In Exercises 6.3.17-6.3.22 find \(a_{0}, ... a_{N}\) for \(N\) at least \(7\) in the power series \(y=\sum_{n=0}^{\infty} a_{n}(x-x_{0})^{n}\) for the solution of the initial value problem. Take \(x_{0}\) to be the point where the initial conditions are imposed.
17. \((x^2-4)y''-xy'-3y=0,\quad y(0)=-1,\quad y'(0)=2\)
18. \(y''+(x-3)y'+3y=0,\quad y(3)=-2,\quad y'(3)=3\)
19. \((5-6x+3x^2)y''+(x-1)y'+12y=0,\quad y(1)=-1,\quad y'(1)=1\)
20. \((4x^2-24x+37)y''+y=0,\quad y(3)=4,\quad y'(3)=-6\)
21. \((x^2-8x+14)y''-8(x-4)y'+20y=0,\quad y(4)=3,\quad y'(4)=-4\)
22. \((2x^2+4x+5)y''-20(x+1)y'+60y=0,\quad y(-1)=3,\quad y'(-1)=-3\)
Q6.3.5
23.
- Find a power series in \(x\) for the general solution of \[(1+x^2)y''+4xy'+2y=0. \tag{A}\]
- Use (a) and the formula \[{1\over1-r}=1+r+r^2+\cdots+r^n+\cdots \quad(-1<r<1)\nonumber \] for the sum of a geometric series to find a closed form expression for the general solution of (A) on \((-1,1)\).
- Show that the expression obtained in (b) is actually the general solution of of (A) on \((-\infty,\infty)\).