What is the general solution for 2sinh(x)-2cosh(x)-2=0 ?

The general solution for 2sinh(x)-2cosh(x)-2=0 is No Solution for x\in\mathbb{R}

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2sinh(x)-2cosh(x)-2=0

Solution

2 sinh (x) - 2 cosh (x) - 2 = 0

Solution

No Solution for x \in R

Solution steps

2 sinh (x) - 2 cosh (x) - 2 = 0

Rewrite using trig identities

2 sinh (x) - 2 cosh (x) - 2 = 0

Use the Hyperbolic identity:

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

2 \cdot \frac{e ^{x} - e ^{- x}}{2} - 2 cosh (x) - 2 = 0

Use the Hyperbolic identity:

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

2 \cdot \frac{e ^{x} - e ^{- x}}{2} - 2 \cdot \frac{e ^{x} + e ^{- x}}{2} - 2 = 0

2 \cdot \frac{e ^{x} - e ^{- x}}{2} - 2 \cdot \frac{e ^{x} + e ^{- x}}{2} - 2 = 0

2 \cdot \frac{e ^{x} - e ^{- x}}{2} - 2 \cdot \frac{e ^{x} + e ^{- x}}{2} - 2 = 0 :

No Solution for

x \in R

2 \cdot \frac{e ^{x} - e ^{- x}}{2} - 2 \cdot \frac{e ^{x} + e ^{- x}}{2} - 2 = 0

Apply exponent rules

2 \cdot \frac{e ^{x} - e ^{- x}}{2} - 2 \cdot \frac{e ^{x} + e ^{- x}}{2} - 2 = 0

Apply exponent rule:

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

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

2 \cdot \frac{e ^{x} - ( e ^{x} ) ^{- 1}}{2} - 2 \cdot \frac{e ^{x} + ( e ^{x} ) ^{- 1}}{2} - 2 = 0

2 \cdot \frac{e ^{x} - ( e ^{x} ) ^{- 1}}{2} - 2 \cdot \frac{e ^{x} + ( e ^{x} ) ^{- 1}}{2} - 2 = 0

Rewrite the equation with

e^{x} = u

2 \cdot \frac{u - ( u ) ^{- 1}}{2} - 2 \cdot \frac{u + ( u ) ^{- 1}}{2} - 2 = 0

Solve

2 \cdot \frac{u - u ^{- 1}}{2} - 2 \cdot \frac{u + u ^{- 1}}{2} - 2 = 0 : u = - 1

2 \cdot \frac{u - u ^{- 1}}{2} - 2 \cdot \frac{u + u ^{- 1}}{2} - 2 = 0

Refine

u - \frac{1}{u} - (u + \frac{1}{u}) - 2 = 0

Simplify

- (u + \frac{1}{u}) : - u - \frac{1}{u}

- (u + \frac{1}{u})

Distribute parentheses

= - (u) - (\frac{1}{u})

Apply minus-plus rules

+ (- a) = - a

= - u - \frac{1}{u}

u - \frac{1}{u} - u - \frac{1}{u} - 2 = 0

Multiply both sides by

u

u - \frac{1}{u} - u - \frac{1}{u} - 2 = 0

Multiply both sides by

u

uu - \frac{1}{u} u - uu - \frac{1}{u} u - 2 u = 0 \cdot u

Simplify

uu - \frac{1}{u} u - uu - \frac{1}{u} u - 2 u = 0 \cdot u

Simplify

uu : u^{2}

uu

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

uu = u^{1 + 1}

= u^{1 + 1}

Add the numbers:

1 + 1 = 2

= u^{2}

Simplify

- \frac{1}{u} u : - 1

- \frac{1}{u} u

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= - \frac{1 \cdot u}{u}

Cancel the common factor:

u

= - 1

Simplify

- uu : - u^{2}

- uu

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

uu = u^{1 + 1}

= - u^{1 + 1}

Add the numbers:

1 + 1 = 2

= - u^{2}

Simplify

- \frac{1}{u} u : - 1

- \frac{1}{u} u

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= - \frac{1 \cdot u}{u}

Cancel the common factor:

u

= - 1

Simplify

0 \cdot u : 0

0 \cdot u

Apply rule

0 \cdot a = 0

= 0

u^{2} - 1 - u^{2} - 1 - 2 u = 0

Simplify

u^{2} - 1 - u^{2} - 1 - 2 u : - 2 u - 2

u^{2} - 1 - u^{2} - 1 - 2 u

Group like terms

= u^{2} - u^{2} - 2 u - 1 - 1

Add similar elements:

u^{2} - u^{2} = 0

= - 2 u - 1 - 1

Subtract the numbers:

- 1 - 1 = - 2

= - 2 u - 2

- 2 u - 2 = 0

- 2 u - 2 = 0

- 2 u - 2 = 0

Move

2

to the right side

- 2 u - 2 = 0

Add

2

to both sides

- 2 u - 2 + 2 = 0 + 2

Simplify

- 2 u = 2

- 2 u = 2

Divide both sides by

- 2

- 2 u = 2

Divide both sides by

- 2

\frac{- 2 u}{- 2} = \frac{2}{- 2}

Simplify

u = - 1

u = - 1

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

2 \frac{u - u ^{- 1}}{2} - 2 \frac{u + u ^{- 1}}{2} - 2

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = - 1

u = - 1

Substitute back

u = e^{x},

solve for

x

Solve

e^{x} = - 1 :

No Solution for

x \in R

e^{x} = - 1

a^{f (x)}

cannot be zero or negative for

x \in R

No Solution for x \in R

No Solution for x \in R

No Solution for x \in R

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sin(x)= 1/2 ,0<x< pi/2

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Frequently Asked Questions (FAQ)

What is the general solution for 2sinh(x)-2cosh(x)-2=0 ?
The general solution for 2sinh(x)-2cosh(x)-2=0 is No Solution for x\in\mathbb{R}