... \#4(H)¹

(H) denotes a problem you should do by hand (you may use a table of integrals); you may use Maple to check your answers. Other problems may be done by hand or with Maple.

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...,10(H),18,24,26,*35²

Asterisks denote unusually hard problems.

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...)³

The answer in the back of the book is wrong. The correct answer is x = 13x₁ + 40x₂ + 3x₃ - 12x₄.

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... \#6(H),8(H),26,*34⁴

answer is $\approx$ 21.4663 pounds

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...,*36⁵

answer is 4, 10, and 4 pounds

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... \#9(H)⁶

From now on all problems marked (H) may be done using a table of Laplace transforms.

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... \#6(H)⁷

answer is in the back of the book

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... \#17(H),21(H),27(H)⁸

Hint: differentiate

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... \#2(H),21(H),*28⁹

answer is in the back of the book

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... \#9¹⁰

answer is $\pi$

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...,13¹¹

answer is ${\frac{{1}}{{2}}}$ + ${\frac{{2}}{{\pi}}}$$\left(\vphantom{ \sin t + \frac{1}{3} \sin 3t + \frac{1}{5} \sin 5t + \dots }\right.$sin t + ${\frac{{1}}{{3}}}$sin 3t + ${\frac{{1}}{{5}}}$sin 5t + ...$\left.\vphantom{ \sin t + \frac{1}{3} \sin 3t + \frac{1}{5} \sin 5t + \dots }\right)$

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...,19¹²

answer is a₀ = ${\frac{{\pi}}{{2}}}$; a_n = ${\frac{{1-(-1)^n}}{{n^2 \pi}}}$ for n $\geq$ 1; b_n = - ${\frac{{1}}{{n}}}$ for n $\geq$ 1

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... sketches.¹³

You may also want to use graphs to check your answer.

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... \#3¹⁴

answer is ${\frac{{1}}{{2}}}$ - ${\frac{{6}}{{\pi}}}$$\left(\vphantom{ \sin \frac{t}{2} + \frac{1}{3} \sin \frac{3t}{2} + \frac{1}{5} \sin \frac{5t}{2} + \dots }\right.$sin${\frac{{t}}{{2}}}$ + ${\frac{{1}}{{3}}}$sin${\frac{{3t}}{{2}}}$ + ${\frac{{1}}{{5}}}$sin${\frac{{5t}}{{2}}}$ + ...$\left.\vphantom{ \sin \frac{t}{2} + \frac{1}{3} \sin \frac{3t}{2} + \frac{1}{5} \sin \frac{5t}{2} + \dots }\right)$

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... sketches.¹⁵

You may also want to use graphs to check your answer.

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... \#1¹⁶

answer is a₀ = 2, a_n = 0 for n $\geq$ 1; b_n = ${\frac{{4}}{{n \pi}}}$ for n odd, b_n = 0 for n even.

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... 7¹⁷

answer is a₀ = ${\frac{{\pi^2}}{{3}}}$, a_n = - 2 ^. ${\frac{{1+(-1)^n}}{{n^2}}}$ for n $\geq$ 1; b_n = 4 ^. ${\frac{{1- (-1)^n}}{{\pi n^3}}}$ for n $\geq$ 1

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... sketches.¹⁸

You may also want to use graphs to check your answer.

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... \#12¹⁹

answer is - ${\frac{{4}}{{\pi}}}$$\left(\vphantom{ \frac{\sin t}{5} + \frac{\sin 3t}{39} + \frac{\sin 5t}{145} + \frac{\sin 7t}{371} + \cdots}\right.$${\frac{{\sin t}}{{5}}}$ + ${\frac{{\sin 3t}}{{39}}}$ + ${\frac{{\sin 5t}}{{145}}}$ + ${\frac{{\sin 7t}}{{371}}}$ + ^... $\left.\vphantom{ \frac{\sin t}{5} + \frac{\sin 3t}{39} + \frac{\sin 5t}{145} + \frac{\sin 7t}{371} + \cdots}\right)$

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...,13²⁰

answer is $$\sum_{{n=1}}^{{\infty}}$$$${\frac{{ 2 \cdot (-1)^n}}{{n\pi (n^2\pi^2-1)}}}$$sin n$$\pi$$t

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... \#11²¹

Don't forget to extend f (t) to an odd function!

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... \#1²²

answer is x_sp(t) = $${\frac{{12}}{{\pi}}}$$$$\sum_{{n \text{ odd}}}^{}$$$${\frac{{1}}{{n(5-n^2)}}}$$sin nt

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...,7²³

answer is yes

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...,8²⁴

answer is no

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...,9²⁵

answer is no

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...)²⁶

answer to (b) is 2.98 in

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