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3 years ago

1. The number a is 4/5 of the number b. What part of number a is number b?

1 answer:

7 0

Writing some equations out might help!

1) You know that a is 4/5 of b. This can be written as: $a = \frac{4}{5} b$ .
To find what b is equal to, multiply both sides by 5/4: $\frac{5}{4}a = b$ .
A is 5/4 B.

2) You know that a is 1/5 smaller than b. This can be written as: $a = b - \frac{1}{5}$ .
To find b, just add 1/5 to both sides: $a + \frac{1}{5} = b$ .
B is 1/5 bigger than a.

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From the homogeneous part of the ODE, we can get two fundamental solutions. The characteristic equation is

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which has roots at $r=1\pm i$ . This admits the two fundamental solutions

$y_1=e^x\cos x$
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The particular solution is easiest to obtain via variation of parameters. We're looking for a solution of the form

$y_p=u_1y_1+u_2y_2$

where

$u_1=-\displaystyle\frac13\int\frac{y_2e^x\sec x}{W(y_1,y_2)}\,\mathrm dx$
$u_2=\displaystyle\frac13\int\frac{y_1e^x\sec x}{W(y_1,y_2)}\,\mathrm dx$

and $W(y_1,y_2)$ is the Wronskian of the fundamental solutions. We have

$W(e^x\cos x,e^x\sin x)=\begin{vmatrix}e^x\cos x&e^x\sin x\\e^x(\cos x-\sin x)&e^x(\cos x+\sin x)\end{vmatrix}=e^{2x}$

and so

$u_1=-\displaystyle\frac13\int\frac{e^{2x}\sin x\sec x}{e^{2x}}\,\mathrm dx=-\int\tan x\,\mathrm dx$
$u_1=\dfrac13\ln|\cos x|$

$u_2=\displaystyle\frac13\int\frac{e^{2x}\cos x\sec x}{e^{2x}}\,\mathrm dx=\int\mathrm dx$
$u_2=\dfrac13x$

Therefore the particular solution is

$y_p=\dfrac13e^x\cos x\ln|\cos x|+\dfrac13xe^x\sin x$

so that the general solution to the ODE is

$y=C_1e^x\cos x+C_2e^x\sin x+\dfrac13e^x\cos x\ln|\cos x|+\dfrac13xe^x\sin x$

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