91Ó°ÊÓ

Question: In Problems $\mathbf{17}\mathbf{-}\mathbf{22}$, solve the initial value problem.

$\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{+}\frac{\mathbf{3}\mathbf{y}}{\mathbf{x}}\mathbf{+}\mathbf{2}\mathbf{=}\mathbf{3}\mathbf{x}\mathbf{,}\mathbf{}\mathbf{}\mathbf{y}\left(1\right)\mathbf{=}\mathbf{1}$

Question: In Problems 17-22, solve the initial value problem.

$\mathbf{\left(}\mathbf{cosx}\mathbf{\right)}\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{+}\mathbf{ysinx}\mathbf{=}\mathbf{2}{\mathbf{xcos}}^{\mathbf{2}}\mathbf{x}\mathbf{,}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{y}\mathbf{\left(}\frac{\mathbf{Ï€}}{\mathbf{4}}\mathbf{\right)}\mathbf{=}\mathbf{-}\frac{\mathbf{15}\sqrt{\mathbf{2}}{\mathbf{Ï€}}^{\mathbf{2}}}{\mathbf{32}}$

Question: In Problems 17-22, solve the initial value problem.

$\mathbf{\left(}\mathbf{sinx}\mathbf{\right)}\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{+}\mathbf{ycosx}\mathbf{=}\mathbf{xsinx}\mathbf{,}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{y}\mathbf{\left(}\frac{\mathbf{Ï€}}{\mathbf{2}}\mathbf{\right)}\mathbf{=}\mathbf{2}$

Question: Radioactive Decay. In Example 2 assume that the rate at which${\mathbf{RA}}_{\mathbf{1}}$ decays into ${\mathbf{RA}}_{\mathbf{2}}$is $\mathbf{40}{\mathbf{e}}^{\mathbf{-}\mathbf{20}\mathbf{t}}\mathbf{}\mathbf{kg}\mathbf{/}\mathbf{sec}$and the decay constant for ${\mathbf{RA}}_{\mathbf{2}}$is $\mathbf{k}\mathbf{=}\mathbf{5}\mathbf{/}\mathbf{sec}$. Find the mass of ${\mathbf{RA}}_{\mathbf{2}}$for $\mathbf{t}\mathbf{â‰\yen }\mathbf{0}$if initially $\mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{10}\mathbf{}\mathbf{kg}$.

Question: In Example 2 the decay constant for isotope ${\mathbf{RA}}_{\mathbf{1}}$was $\mathbf{k}\mathbf{=}\mathbf{10}\mathbf{/}\mathbf{sec}$, which expresses itself in the exponent of the rate term $\mathbf{50}{\mathbf{e}}^{\mathbf{-}\mathbf{10}\mathbf{t}}\mathbf{kg}\mathbf{/}\mathbf{sec}$. When the decay constant for ${\mathbf{RA}}_{\mathbf{2}}$is $\mathbf{k}\mathbf{=}\mathbf{5}\mathbf{/}\mathbf{sec}$, we see that in formula (14) for y the term$\left(\frac{185}{4}\right){\mathbf{e}}^{\mathbf{-}\mathbf{2}\mathbf{t}}$ eventually dominates (has greater magnitude for t large).

1. Redo Example 2 taking $\mathbf{k}\mathbf{=}\mathbf{20}\mathbf{/}\mathbf{sec}$. Now which term in the solution eventually dominates?
2. Redo Example 2 taking $\mathbf{k}\mathbf{=}\mathbf{10}\mathbf{/}\mathbf{sec}$.

Question: (a) Using definite integration, show that the solution to the initial value problem $\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{+}\mathbf{2}\mathbf{xy}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{y}\mathbf{\left(}\mathbf{2}\mathbf{\right)}\mathbf{=}\mathbf{1}$can be expressed as $\mathbf{y}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}{\mathbf{e}}^{\mathbf{-}{\mathbf{x}}^{\mathbf{2}}}\left(e^4+{∫}_2^x{e}^{t^2}\mathrm{dt}\right)$.

(b) Use numerical integration (such as Simpson’s rule, Appendix C) to approximate the solution at $\mathbf{x}\mathbf{=}\mathbf{3}$.

Question: Use numerical integration (such as Simpson’s rule, Appendix C) to approximate the solution, at x = 1, to the initial value problem

$\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{+}\frac{\mathbf{sin}\mathbf{2}\mathbf{x}}{\mathbf{2}\mathbf{(}\mathbf{1}\mathbf{+}{\mathbf{sin}}^{\mathbf{2}}\mathbf{x}\mathbf{)}}\mathbf{y}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{0}$

Ensure your approximation is accurate to three decimal places.

Question: In Problems 7-16, obtain the general solution to the equation.

$\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{-}\mathbf{y}\mathbf{-}{\mathbf{e}}^{\mathbf{3}\mathbf{x}}\mathbf{=}\mathbf{0}$

Question: In Problems 7-16, obtain the general solution to the equation.

$\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{=}\frac{\mathbf{y}}{\mathbf{x}}\mathbf{+}\mathbf{2}\mathbf{x}\mathbf{+}\mathbf{1}$

Question: In Problems 7-16, obtain the general solution to the equation.

$\frac{\mathbf{dr}}{\mathbf{»åθ}}\mathbf{+}\mathbf{³Ù²¹²Ôθ}\mathbf{=}\mathbf{²õ\pm ð³¦Î¸}$