What is the general solution for sinh(3x)-3sinh(x)=0 ?

The general solution for sinh(3x)-3sinh(x)=0 is x=0

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sinh(3x)-3sinh(x)=0

Solution

sinh (3 x) - 3 sinh (x) = 0

Solution

x = 0

Degrees

x = 0^{\circ}

Solution steps

sinh (3 x) - 3 sinh (x) = 0

Rewrite using trig identities

sinh (3 x) - 3 sinh (x) = 0

Use the Hyperbolic identity:

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

\frac{e ^{3 x} - e ^{- 3 x}}{2} - 3 \cdot \frac{e ^{x} - e ^{- x}}{2} = 0

\frac{e ^{3 x} - e ^{- 3 x}}{2} - 3 \cdot \frac{e ^{x} - e ^{- x}}{2} = 0

\frac{e ^{3 x} - e ^{- 3 x}}{2} - 3 \cdot \frac{e ^{x} - e ^{- x}}{2} = 0 : x = 0

\frac{e ^{3 x} - e ^{- 3 x}}{2} - 3 \cdot \frac{e ^{x} - e ^{- x}}{2} = 0

Multiply both sides by

2

\frac{e ^{3 x} - e ^{- 3 x}}{2} \cdot 2 - 3 \cdot \frac{e ^{x} - e ^{- x}}{2} \cdot 2 = 0 \cdot 2

Simplify

e^{3 x} - e^{- 3 x} - 3 (e^{x} - e^{- x}) = 0

Apply exponent rules

e^{3 x} - e^{- 3 x} - 3 (e^{x} - e^{- x}) = 0

Apply exponent rule:

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

e^{3 x} = (e^{x})^{3}, e^{- 3 x} = (e^{x})^{- 3}, e^{- x} = (e^{x})^{- 1}

(e^{x})^{3} - (e^{x})^{- 3} - 3 (e^{x} - (e^{x})^{- 1}) = 0

(e^{x})^{3} - (e^{x})^{- 3} - 3 (e^{x} - (e^{x})^{- 1}) = 0

Rewrite the equation with

e^{x} = u

(u)^{3} - (u)^{- 3} - 3 (u - (u)^{- 1}) = 0

Solve

u^{3} - u^{- 3} - 3 (u - u^{- 1}) = 0 : u = 1, u = - 1

u^{3} - u^{- 3} - 3 (u - u^{- 1}) = 0

Refine

u^{3} - \frac{1}{u ^{3}} - 3 (u - \frac{1}{u}) = 0

Multiply both sides by

u^{3}

u^{3} - \frac{1}{u ^{3}} - 3 (u - \frac{1}{u}) = 0

Multiply both sides by

u^{3}

u^{3} u^{3} - \frac{1}{u ^{3}} u^{3} - 3 (u - \frac{1}{u}) u^{3} = 0 \cdot u^{3}

Simplify

u^{3} u^{3} - \frac{1}{u ^{3}} u^{3} - 3 (u - \frac{1}{u}) u^{3} = 0 \cdot u^{3}

Simplify

u^{3} u^{3} : u^{6}

u^{3} u^{3}

Apply exponent rule:

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

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

= u^{3 + 3}

Add the numbers:

3 + 3 = 6

= u^{6}

Simplify

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

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

Multiply fractions:

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

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

Cancel the common factor:

u^{3}

= - 1

Simplify

0 \cdot u^{3} : 0

0 \cdot u^{3}

Apply rule

0 \cdot a = 0

= 0

u^{6} - 1 - 3 (u - \frac{1}{u}) u^{3} = 0

u^{6} - 1 - 3 (u - \frac{1}{u}) u^{3} = 0

u^{6} - 1 - 3 (u - \frac{1}{u}) u^{3} = 0

Expand

u^{6} - 1 - 3 (u - \frac{1}{u}) u^{3} : u^{6} - 1 - 3 u^{4} + 3 u^{2}

u^{6} - 1 - 3 (u - \frac{1}{u}) u^{3}

= u^{6} - 1 - 3 u^{3} (u - \frac{1}{u})

Expand

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

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

Apply the distributive law:

a (b - c) = ab - a c

a = - 3 u^{3}, b = u, c = \frac{1}{u}

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

Apply minus-plus rules

- (- a) = a

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

Simplify

- 3 u^{3} u + 3 \cdot \frac{1}{u} u^{3} : - 3 u^{4} + 3 u^{2}

- 3 u^{3} u + 3 \cdot \frac{1}{u} u^{3}

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

3 u^{3} u

Apply exponent rule:

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

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

= 3 u^{3 + 1}

Add the numbers:

3 + 1 = 4

= 3 u^{4}

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

3 \cdot \frac{1}{u} u^{3}

Multiply fractions:

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

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

Multiply the numbers:

1 \cdot 3 = 3

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

Cancel the common factor:

u

= 3 u^{2}

= - 3 u^{4} + 3 u^{2}

= - 3 u^{4} + 3 u^{2}

= u^{6} - 1 - 3 u^{4} + 3 u^{2}

u^{6} - 1 - 3 u^{4} + 3 u^{2} = 0

Solve

u^{6} - 1 - 3 u^{4} + 3 u^{2} = 0 : u = 1, u = - 1

u^{6} - 1 - 3 u^{4} + 3 u^{2} = 0

Write in the standard form

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

u^{6} - 3 u^{4} + 3 u^{2} - 1 = 0

Rewrite the equation with

v = u^{2}, v^{2} = u^{4}

and

v^{3} = u^{6}

v^{3} - 3 v^{2} + 3 v - 1 = 0

Solve

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

v^{3} - 3 v^{2} + 3 v - 1 = 0

Factor

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

v^{3} - 3 v^{2} + 3 v - 1

Apply cube of difference rule:

a^{3} - 3 a^{2} b + 3 a b^{2} - b^{3} = (a - b)^{3}

a = v, b = 1

= (v - 1)^{3}

(v - 1)^{3} = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

v - 1 = 0

Solve

v - 1 = 0 : v = 1

v - 1 = 0

Move

1

to the right side

v - 1 = 0

Add

1

to both sides

v - 1 + 1 = 0 + 1

Simplify

v = 1

v = 1

The solution is

v = 1

v = 1

Substitute back

v = u^{2},

solve for

u

Solve

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

u^{2} = 1

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = 1, u = - 1

1 = 1

1

Apply radical rule:

1 = 1

= 1

- 1 = - 1

- 1

Apply radical rule:

1 = 1

1 = 1

= - 1

u = 1, u = - 1

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

and compare to zero

Solve

u^{3} = 0 : u = 0

u^{3} = 0

Apply rule

x^{n} = 0 \Rightarrow x = 0

u = 0

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(3x)-3sinh(x)=0 ?
The general solution for sinh(3x)-3sinh(x)=0 is x=0