X - ay - a - yay - a - y To show y - 12log a 1 - x1 - x

x= a ^ y a ^ y a ^ y + a ^ y = a ^ 2y 1 a ^ 2y +1 1+x 1 x = 2 a ^ 2y 2 = a ^ 2y taking log on base a on both sides log _ a ( 1+ ...

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Byju's Answer

Standard XII

Mathematics

Particular Solution of a Differential Equation

x = ay - a - ...

Question

$x = \frac{a^{y} - a^{- y}}{a^{y} + a^{- y}}$
To show $y = \frac{1}{2} {log}_{a} (\frac{1 + x}{1 - x})$

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Solution

$x = \frac{a^{y} - a^{- y}}{a^{y} + a^{- y}} = \frac{a^{2 y} - 1}{a^{2 y} + 1}$

$\frac{1 + x}{1 - x} = \frac{2 a^{2 y}}{2} = a^{2 y}$

taking log on base $a$ on both sides

${log}_{a} (\frac{1 + x}{1 - x}) = 2 y \Rightarrow y = \frac{1}{2} {log}_{a} (\frac{1 + x}{1 - x})$

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Similar questions

Q. If x=

\frac{a^{y} - a^{- y}}{a^{y} + a^{- y}}

To show

y = \frac{1}{2} {log}_{a} (\frac{1 + x}{1 - x})

Q. Of the following:
(1)

a (x - y) = a x - a y

(2)

a^{x - y} = a^{x} - a^{y}

(3)

log (x - y) = log x - log y

(4)

log x / log y = log x - log y

(5)

a (x y) = a x \cdot a y

Q. Evaluate

l i m y \to 0 ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ l i m x \to \infty \frac{e x p [x ℓ n [1 + \frac{a y}{x}]] - e x p [x ℓ n [1 + \frac{b y}{x}]]}{y} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

Q. If

x = c y + b z, y = c x + a z, z = b x + a y

the value of

a^{2} + b^{2} - 1

The integrating factor (IF) of the differential equation
$(1 - y^{2}) \frac{d y}{d x} + y x = a y (- 1 < y < 1)$ is
(a) $\frac{1}{y^{2} - 1}$
(b) $\frac{1}{\sqrt{y^{2} - 1}}$
(c) $\frac{1}{1 - y^{2}}$
(d) $\frac{1}{\sqrt{1 - y^{2}}}$

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x=ay−a−yay+a−y To show y=12loga(1+x1−x)

x=ay−a−yay+a−y=a2y−1a2y+11+x1−x=2a2y2=a2ytaking log on base a on both sidesloga(1+x1−x)=2y⇒y=12loga(1+x1−x)

$x = \frac{a^{y} - a^{- y}}{a^{y} + a^{- y}}$
To show $y = \frac{1}{2} {log}_{a} (\frac{1 + x}{1 - x})$

$x = \frac{a^{y} - a^{- y}}{a^{y} + a^{- y}} = \frac{a^{2 y} - 1}{a^{2 y} + 1}$

$\frac{1 + x}{1 - x} = \frac{2 a^{2 y}}{2} = a^{2 y}$

taking log on base $a$ on both sides

${log}_{a} (\frac{1 + x}{1 - x}) = 2 y \Rightarrow y = \frac{1}{2} {log}_{a} (\frac{1 + x}{1 - x})$