What is the general solution for solvefor u,x=4sinh(u) ?

The general solution for solvefor u,x=4sinh(u) is u=ln((x+sqrt(x^2+16))/4),u=ln((x-sqrt(x^2+16))/4)

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solvefor u,x=4sinh(u)

Solution

solve for

u, x = 4 sinh (u)

Solution

u = ln (\frac{x + x ^{2} + 16}{4}), u = ln (\frac{x - x ^{2} + 16}{4})

Solution steps

x = 4 sinh (u)

Switch sides

4 sinh (u) = x

Rewrite using trig identities

4 sinh (u) = x

Use the Hyperbolic identity:

sinh (x) = \frac{e ^{x} - e ^{- x}}{2}

4 \cdot \frac{e ^{u} - e ^{- u}}{2} = x

4 \cdot \frac{e ^{u} - e ^{- u}}{2} = x

4 \cdot \frac{e ^{u} - e ^{- u}}{2} = x : u = ln (\frac{x + x ^{2} + 16}{4}), u = ln (\frac{x - x ^{2} + 16}{4})

4 \cdot \frac{e ^{u} - e ^{- u}}{2} = x

Apply exponent rules

4 \cdot \frac{e ^{u} - e ^{- u}}{2} = x

Apply exponent rule:

a^{b c} = (a^{b})^{c}

e^{- u} = (e^{u})^{- 1}

4 \cdot \frac{e ^{u} - ( e ^{u} ) ^{- 1}}{2} = x

4 \cdot \frac{e ^{u} - ( e ^{u} ) ^{- 1}}{2} = x

Rewrite the equation with

e^{u} = v

4 \cdot \frac{v - v ^{- 1}}{2} = x

Solve

4 \cdot \frac{v - v ^{- 1}}{2} = x : v = \frac{x + x ^{2} + 16}{4}, v = \frac{x - x ^{2} + 16}{4}

4 \cdot \frac{v - v ^{- 1}}{2} = x

Refine

\frac{2 ( v ^{2} - 1 )}{v} = x

Multiply both sides by

v

\frac{2 ( v ^{2} - 1 )}{v} = x

Multiply both sides by

v

\frac{2 ( v ^{2} - 1 )}{v} v = xv

Simplify

2 (v^{2} - 1) = xv

2 (v^{2} - 1) = xv

Solve

2 (v^{2} - 1) = xv : v = \frac{x + x ^{2} + 16}{4}, v = \frac{x - x ^{2} + 16}{4}

2 (v^{2} - 1) = xv

Expand

2 (v^{2} - 1) : 2 v^{2} - 2

2 (v^{2} - 1)

Apply the distributive law:

a (b - c) = ab - a c

a = 2, b = v^{2}, c = 1

= 2 v^{2} - 2 \cdot 1

Multiply the numbers:

2 \cdot 1 = 2

= 2 v^{2} - 2

2 v^{2} - 2 = xv

Move

xv

to the left side

2 v^{2} - 2 = xv

Subtract

xv

from both sides

2 v^{2} - 2 - xv = xv - xv

Simplify

2 v^{2} - 2 - xv = 0

2 v^{2} - 2 - xv = 0

Write in the standard form

a x^{2} + b x + c = 0

2 v^{2} - xv - 2 = 0

Solve with the quadratic formula

2 v^{2} - xv - 2 = 0

Quadratic Equation Formula:

For

a = 2, b = - x, c = - 2

v_{1, 2} = \frac{- ( - x ) \pm ( - x ) ^{2} - 4 \cdot 2 ( - 2 )}{2 \cdot 2}

v_{1, 2} = \frac{- ( - x ) \pm ( - x ) ^{2} - 4 \cdot 2 ( - 2 )}{2 \cdot 2}

Simplify

(- x)^{2} - 4 \cdot 2 (- 2) : x^{2} + 16

(- x)^{2} - 4 \cdot 2 (- 2)

Apply rule

- (- a) = a

= (- x)^{2} + 4 \cdot 2 \cdot 2

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- x)^{2} = x^{2}

= x^{2} + 4 \cdot 2 \cdot 2

Multiply the numbers:

4 \cdot 2 \cdot 2 = 16

= x^{2} + 16

v_{1, 2} = \frac{- ( - x ) \pm x ^{2} + 16}{2 \cdot 2}

Separate the solutions

v_{1} = \frac{- ( - x ) + x ^{2} + 16}{2 \cdot 2}, v_{2} = \frac{- ( - x ) - x ^{2} + 16}{2 \cdot 2}

v = \frac{- ( - x ) + x ^{2} + 16}{2 \cdot 2} : \frac{x + x ^{2} + 16}{4}

\frac{- ( - x ) + x ^{2} + 16}{2 \cdot 2}

Apply rule

- (- a) = a

= \frac{x + x ^{2} + 16}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{x + x ^{2} + 16}{4}

v = \frac{- ( - x ) - x ^{2} + 16}{2 \cdot 2} : \frac{x - x ^{2} + 16}{4}

\frac{- ( - x ) - x ^{2} + 16}{2 \cdot 2}

Apply rule

- (- a) = a

= \frac{x - x ^{2} + 16}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{x - x ^{2} + 16}{4}

The solutions to the quadratic equation are:

v = \frac{x + x ^{2} + 16}{4}, v = \frac{x - x ^{2} + 16}{4}

v = \frac{x + x ^{2} + 16}{4}, v = \frac{x - x ^{2} + 16}{4}

v = \frac{x + x ^{2} + 16}{4}, v = \frac{x - x ^{2} + 16}{4}

Substitute back

v = e^{u},

solve for

u

Solve

e^{u} = \frac{x + x ^{2} + 16}{4} : u = ln (\frac{x + x ^{2} + 16}{4})

e^{u} = \frac{x + x ^{2} + 16}{4}

Apply exponent rules

e^{u} = \frac{x + x ^{2} + 16}{4}

f (x) = g (x)

, then

ln (f (x)) = ln (g (x))

ln (e^{u}) = ln (\frac{x + x ^{2} + 16}{4})

Apply log rule:

ln (e^{a}) = a

ln (e^{u}) = u

u = ln (\frac{x + x ^{2} + 16}{4})

u = ln (\frac{x + x ^{2} + 16}{4})

Solve

e^{u} = \frac{x - x ^{2} + 16}{4} : u = ln (\frac{x - x ^{2} + 16}{4})

e^{u} = \frac{x - x ^{2} + 16}{4}

Apply exponent rules

e^{u} = \frac{x - x ^{2} + 16}{4}

f (x) = g (x)

, then

ln (f (x)) = ln (g (x))

ln (e^{u}) = ln (\frac{x - x ^{2} + 16}{4})

Apply log rule:

ln (e^{a}) = a

ln (e^{u}) = u

u = ln (\frac{x - x ^{2} + 16}{4})

u = ln (\frac{x - x ^{2} + 16}{4})

u = ln (\frac{x + x ^{2} + 16}{4}), u = ln (\frac{x - x ^{2} + 16}{4})

u = ln (\frac{x + x ^{2} + 16}{4}), u = ln (\frac{x - x ^{2} + 16}{4})

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Frequently Asked Questions (FAQ)

What is the general solution for solvefor u,x=4sinh(u) ?
The general solution for solvefor u,x=4sinh(u) is u=ln((x+sqrt(x^2+16))/4),u=ln((x-sqrt(x^2+16))/4)