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chap 6 differential equation
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Chapter 6 - In Review
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2. False; $y = x$ is a solution that is analytic at $x = 0$.
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5. The interval of convergence is centered at 4. Since the series converges at —2, it converges at least in the interval [—2.10). Since it diverges at 13, it converges at most on the interval [—5,13). Thus, at —7 it does not converge, at 0 and 7 it does converge, and at 10 and 11 it might converge.
6. we have $f(x)=\dfrac{sin\, x}{cos\, x}=\dfrac{x-\dfrac{x^3}{6}+\dfrac{x^5}{120}-\cdots}{1-\dfrac{x^2}{2}+\dfrac{x^4}{24}-\cdots}=x+\dfrac{x^3}{3}+\farc{2x^5}{15}+\cdots$.
7. The differential equation $x^3 — x^2)y" + y' + y = 0$ has a regular singular point at $x = 1$ and has an irregular singular point at $x = 0$.
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17. The singular point of $(1 — 2\, sin \,x)y" + xy = 0$ closest to $x = 0$ is $\dfrac{\pi}{6}$. Hence a lower bound is $\dfrac{\pi}{6}$.
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19. Writing the differential equation in the form $y’’+\left(\dfrac{1-cos\,x}{x}\right)y’+xy=0$, and noting that $\dfrac{1 — cos\,x}{x}=\dfrac{x}{2}-\dfrac{x^3}{24}+\dfrac{x^5}{720}-\cdots$ is analytic at $x = 0$, we conclude that $x = 0$ is an ordinary point, of the differential equation.
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