A.9.1: Section 9.1 Answers

2. \(y=2x^{2}-3x^{3}+\frac{1}{x}\)

3. \(y=2e^{x}+3e^{-x}-e^{2x}+e^{-3x}\)

4. \(y_{i}=\frac{(x-x_{0})^{i-1}}{(i-1)!},\quad 1\leq i\leq n\)

5.

b. \(y_{1}=-\frac{1}{2}x^{3}+x^{2}+\frac{1}{2x},\quad y_{2}=\frac{1}{3}x^{2}-\frac{1}{3x},\quad y_{3}=\frac{1}{4}x^{3}-\frac{1}{3}x^{2}+\frac{1}{12x}\)

c. \(y=k_{0}y_{1}+k_{1}y_{2}+k_{2}y_{3}\)

7. \(2e^{-x^{2}}\)

8. \(\sqrt{2}K\cos x\)

9.

a. \(W(x)=2e^{3x}\)

d. \(y=e^{x}(c_{1}+c_{2}x+c_{3}x^{2})\)

10.

1. \(2\)
2. \(-e^{3x}\)
3. \(4\)
4. \(4/x^{2}\)
5. \(1\)
6. \(2x\)
7. \(2/x^{2}\)
8. \(e^{x}(x^{2}-2x+2)\)
9. \(-240/x^{5}\)
10. \(6e^{2x}(2x-1)\)
11. \(-128x\)

24.

1. \(y'''=0\)
2. \(xy'''-y''-xy'+y=0\)
3. \((2x-3)y'''-2y''-(2x-5)y'=0\)
4. \((x^{2}-2x+2)y'''-x^{2}y''+2xy'-2y=0\)
5. \(x^{3}y'''+x^{2}y''-2xy'+2y=9\)
6. \((3x-1)y'''-(12x-1)y''+9(x+1)y'-9y=0\)
7. \(x^{4}y^{(4)}+5x^{3}y'''-3x^{2}y''-6xy'+6y=0\)
8. \(x^{4}y^{(4)}+3x^{2}y'''-x^{2}y''+2xy'-2y=0\)
9. \((2x-1)y^{(4)}-4xy'''+(5-2x)y''+4xy'-4y=0\)
10. \(xy^{(4)}-y'''-4xy''+4y'=0\)

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