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Alex787 [66]
3 years ago
8

Bob can dig holes and plant 15 trees in 60 minutes. George can accomplish this same task in 120 minutes. If they work together,

how long (minutes) will it take Bob and George to plant the 15 trees? minutes.
Mathematics
2 answers:
Elis [28]3 years ago
8 0

Answer:

40 Minutes

Step-by-step explanation:

yan [13]3 years ago
7 0

Answer:

0

Step-by-step explanation:

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Find the form of the general solution of y^(4)(x) - n^2y''(x)=g(x)
Dennis_Churaev [7]

The differential equation

y^{(4)}-n^2y'' = g(x)

has characteristic equation

<em>r</em> ⁴ - <em>n </em>² <em>r</em> ² = <em>r</em> ² (<em>r</em> ² - <em>n </em>²) = <em>r</em> ² (<em>r</em> - <em>n</em>) (<em>r</em> + <em>n</em>) = 0

with roots <em>r</em> = 0 (multiplicity 2), <em>r</em> = -1, and <em>r</em> = 1, so the characteristic solution is

y_c=C_1+C_2x+C_3e^{-nx}+C_4e^{nx}

For the non-homogeneous equation, reduce the order by substituting <em>u(x)</em> = <em>y''(x)</em>, so that <em>u''(x)</em> is the 4th derivative of <em>y</em>, and

u''-n^2u = g(x)

Solve for <em>u</em> by using the method of variation of parameters. Note that the characteristic equation now only admits the two exponential solutions found earlier; I denote them by <em>u₁ </em>and <em>u₂</em>. Now we look for a particular solution of the form

u_p = u_1z_1 + u_2z_2

where

\displaystyle z_1(x) = -\int\frac{u_2(x)g(x)}{W(u_1(x),u_2(x))}\,\mathrm dx

\displaystyle z_2(x) = \int\frac{u_1(x)g(x)}{W(u_1(x),u_2(x))}\,\mathrm dx

where <em>W</em> (<em>u₁</em>, <em>u₂</em>) is the Wronskian of <em>u₁ </em>and <em>u₂</em>. We have

W(u_1(x),u_2(x)) = \begin{vmatrix}e^{-nx}&e^{nx}\\-ne^{-nx}&ne^{nx}\end{vmatrix} = 2n

and so

\displaystyle z_1(x) = -\frac1{2n}\int e^{nx}g(x)\,\mathrm dx

\displaystyle z_2(x) = \frac1{2n}\int e^{-nx}g(x)\,\mathrm dx

So we have

\displaystyle u_p = -\frac1{2n}e^{-nx}\int_0^x e^{n\xi}g(\xi)\,\mathrm d\xi + \frac1{2n}e^{nx}\int_0^xe^{-n\xi}g(\xi)\,\mathrm d\xi

and hence

u(x)=C_1e^{-nx}+C_2e^{nx}+u_p(x)

Finally, integrate both sides twice to solve for <em>y</em> :

\displaystyle y(x)=C_1+C_2x+C_3e^{-nx}+C_4e^{nx}+\int_0^x\int_0^\omega u_p(\xi)\,\mathrm d\xi\,\mathrm d\omega

7 0
3 years ago
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