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

The general solution for sinh(x)=2cosh(x)sinh(x) is x=0

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sinh(x)=2cosh(x)sinh(x)

Solution

sinh (x) = 2 cosh (x) sinh (x)

Solution

x = 0

Degrees

x = 0^{\circ}

Solution steps

sinh (x) = 2 cosh (x) sinh (x)

Rewrite using trig identities

sinh (x) = 2 cosh (x) sinh (x)

Use the Hyperbolic identity:

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

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

Use the Hyperbolic identity:

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

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

Use the Hyperbolic identity:

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

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

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

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

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

Multiply both sides by

2

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

Simplify

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

Apply exponent rules

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

Apply exponent rule:

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

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

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

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

Rewrite the equation with

e^{x} = u

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

Solve

u - u^{- 1} = (u + u^{- 1}) (u - u^{- 1}) : u = 1, u = - 1

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

Refine

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

Multiply both sides by

u

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

Multiply both sides by

u

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

Simplify

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

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

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

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

Expand

(u + \frac{1}{u}) (u - \frac{1}{u}) u : u^{3} - \frac{1}{u}

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

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

Expand

(u + \frac{1}{u}) (u - \frac{1}{u}) : u^{2} - \frac{1}{u ^{2}}

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

Apply Difference of Two Squares Formula:

(a + b) (a - b) = a^{2} - b^{2}

a = u, b = \frac{1}{u}

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

(\frac{1}{u})^{2} = \frac{1}{u ^{2}}

(\frac{1}{u})^{2}

Apply exponent rule:

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

= \frac{1 ^{2}}{u ^{2}}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{1}{u ^{2}}

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

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

Expand

u (u^{2} - \frac{1}{u ^{2}}) : u^{3} - \frac{1}{u}

u (u^{2} - \frac{1}{u ^{2}})

Apply the distributive law:

a (b - c) = ab - a c

a = u, b = u^{2}, c = \frac{1}{u ^{2}}

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

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

Simplify

u^{2} u - \frac{1}{u ^{2}} u : u^{3} - \frac{1}{u}

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

u^{2} u = u^{3}

u^{2} u

Apply exponent rule:

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

u^{2} u = u^{2 + 1}

= u^{2 + 1}

Add the numbers:

2 + 1 = 3

= u^{3}

\frac{1}{u ^{2}} u = \frac{1}{u}

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

Multiply fractions:

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

= \frac{1 \cdot u}{u ^{2}}

Multiply:

1 \cdot u = u

= \frac{u}{u ^{2}}

Cancel the common factor:

u

= \frac{1}{u}

= u^{3} - \frac{1}{u}

= u^{3} - \frac{1}{u}

= u^{3} - \frac{1}{u}

u^{2} - 1 = u^{3} - \frac{1}{u}

Multiply both sides by

u

u^{2} - 1 = u^{3} - \frac{1}{u}

Multiply both sides by

u

u^{2} u - 1 \cdot u = u^{3} u - \frac{1}{u} u

Simplify

u^{2} u - 1 \cdot u = u^{3} u - \frac{1}{u} u

Simplify

u^{2} u : u^{3}

u^{2} u

Apply exponent rule:

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

u^{2} u = u^{2 + 1}

= u^{2 + 1}

Add the numbers:

2 + 1 = 3

= u^{3}

Simplify

- 1 \cdot u : - u

- 1 \cdot u

Multiply:

1 \cdot u = u

= - u

Simplify

u^{3} u : u^{4}

u^{3} u

Apply exponent rule:

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

u^{3} u = u^{3 + 1}

= u^{3 + 1}

Add the numbers:

3 + 1 = 4

= u^{4}

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

u^{3} - u = u^{4} - 1

u^{3} - u = u^{4} - 1

u^{3} - u = u^{4} - 1

Solve

u^{3} - u = u^{4} - 1 : u = 1, u = - 1

u^{3} - u = u^{4} - 1

Switch sides

u^{4} - 1 = u^{3} - u

Move

u

to the left side

u^{4} - 1 = u^{3} - u

Add

u

to both sides

u^{4} - 1 + u = u^{3} - u + u

Simplify

u^{4} - 1 + u = u^{3}

u^{4} - 1 + u = u^{3}

Move

u^{3}

to the left side

u^{4} - 1 + u = u^{3}

Subtract

u^{3}

from both sides

u^{4} - 1 + u - u^{3} = u^{3} - u^{3}

Simplify

u^{4} - 1 + u - u^{3} = 0

u^{4} - 1 + u - u^{3} = 0

Write in the standard form

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

u^{4} - u^{3} + u - 1 = 0

Factor

u^{4} - u^{3} + u - 1 : (u - 1) (u + 1) (u^{2} - u + 1)

u^{4} - u^{3} + u - 1

= (u^{4} - u^{3}) + (u - 1)

Factor out

u^{3}

from

u^{4} - u^{3} : u^{3} (u - 1)

u^{4} - u^{3}

Apply exponent rule:

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

u^{4} = u u^{3}

= u u^{3} - u^{3}

Factor out common term

u^{3}

= u^{3} (u - 1)

= (u - 1) + u^{3} (u - 1)

Factor out common term

u - 1

= (u - 1) (u^{3} + 1)

Factor

u^{3} + 1 : (u + 1) (u^{2} - u + 1)

u^{3} + 1

Rewrite

1

1^{3}

= u^{3} + 1^{3}

Apply Sum of Cubes Formula:

x^{3} + y^{3} = (x + y) (x^{2} - x y + y^{2})

u^{3} + 1^{3} = (u + 1) (u^{2} - u + 1)

= (u + 1) (u^{2} - u + 1)

= (u - 1) (u + 1) (u^{2} - u + 1)

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

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

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

Solve

u - 1 = 0 : u = 1

u - 1 = 0

Move

1

to the right side

u - 1 = 0

Add

1

to both sides

u - 1 + 1 = 0 + 1

Simplify

u = 1

u = 1

Solve

u + 1 = 0 : u = - 1

u + 1 = 0

Move

1

to the right side

u + 1 = 0

Subtract

1

from both sides

u + 1 - 1 = 0 - 1

Simplify

u = - 1

u = - 1

Solve

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

No Solution for

u \in R

u^{2} - u + 1 = 0

Discriminant

u^{2} - u + 1 = 0 : - 3

u^{2} - u + 1 = 0

For a quadratic equation of the form

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

the discriminant is

b^{2} - 4 a c

For

a = 1, b = - 1, c = 1 : (- 1)^{2} - 4 \cdot 1 \cdot 1

(- 1)^{2} - 4 \cdot 1 \cdot 1

Expand

(- 1)^{2} - 4 \cdot 1 \cdot 1 : - 3

(- 1)^{2} - 4 \cdot 1 \cdot 1

(- 1)^{2} = 1

(- 1)^{2}

Apply exponent rule:

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

n

is even

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

= 1^{2}

Apply rule

1^{a} = 1

= 1

4 \cdot 1 \cdot 1 = 4

4 \cdot 1 \cdot 1

Multiply the numbers:

4 \cdot 1 \cdot 1 = 4

= 4

= 1 - 4

Subtract the numbers:

1 - 4 = - 3

= - 3

- 3

Discriminant cannot be negative for

u \in R

The solution is

No Solution for u \in R

The solutions are

u = 1, u = - 1

u = 1, u = - 1

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

u - u^{- 1}

and compare to zero

u = 0

Take the denominator(s) of

(u + u^{- 1}) (u - u^{- 1})

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = 1, u = - 1

u = 1, u = - 1

Substitute back

u = e^{x},

solve for

x

Solve

e^{x} = 1 : x = 0

e^{x} = 1

Apply exponent rules

e^{x} = 1

f (x) = g (x)

, then

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

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

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (1)

Simplify

ln (1) : 0

ln (1)

Apply log rule:

lo g_{a} (1) = 0

= 0

x = 0

x = 0

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

x = 0

x = 0

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Frequently Asked Questions (FAQ)

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