What is the general solution for sinh(x)-5cosh(x)=-7 ?

The general solution for sinh(x)-5cosh(x)=-7 is x=-ln(2),x=ln(3)

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sinh(x)-5cosh(x)=-7

Solution

sinh (x) - 5 cosh (x) = - 7

Solution

x = - ln (2), x = ln (3)

Degrees

x = - 39.71440 \dots^{\circ}, x = 62.94584 \dots^{\circ}

Solution steps

sinh (x) - 5 cosh (x) = - 7

Rewrite using trig identities

sinh (x) - 5 cosh (x) = - 7

Use the Hyperbolic identity:

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

\frac{e ^{x} - e ^{- x}}{2} - 5 cosh (x) = - 7

Use the Hyperbolic identity:

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

\frac{e ^{x} - e ^{- x}}{2} - 5 \cdot \frac{e ^{x} + e ^{- x}}{2} = - 7

\frac{e ^{x} - e ^{- x}}{2} - 5 \cdot \frac{e ^{x} + e ^{- x}}{2} = - 7

\frac{e ^{x} - e ^{- x}}{2} - 5 \cdot \frac{e ^{x} + e ^{- x}}{2} = - 7 : x = - ln (2), x = ln (3)

\frac{e ^{x} - e ^{- x}}{2} - 5 \cdot \frac{e ^{x} + e ^{- x}}{2} = - 7

Multiply both sides by

2

\frac{e ^{x} - e ^{- x}}{2} \cdot 2 - 5 \cdot \frac{e ^{x} + e ^{- x}}{2} \cdot 2 = - 7 \cdot 2

Simplify

e^{x} - e^{- x} - 5 (e^{x} + e^{- x}) = - 14

Apply exponent rules

e^{x} - e^{- x} - 5 (e^{x} + e^{- x}) = - 14

Apply exponent rule:

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

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

e^{x} - (e^{x})^{- 1} - 5 (e^{x} + (e^{x})^{- 1}) = - 14

e^{x} - (e^{x})^{- 1} - 5 (e^{x} + (e^{x})^{- 1}) = - 14

Rewrite the equation with

e^{x} = u

u - (u)^{- 1} - 5 (u + (u)^{- 1}) = - 14

Solve

u - u^{- 1} - 5 (u + u^{- 1}) = - 14 : u = \frac{1}{2}, u = 3

u - u^{- 1} - 5 (u + u^{- 1}) = - 14

Refine

u - \frac{1}{u} - 5 (u + \frac{1}{u}) = - 14

Multiply both sides by

u

u - \frac{1}{u} - 5 (u + \frac{1}{u}) = - 14

Multiply both sides by

u

uu - \frac{1}{u} u - 5 (u + \frac{1}{u}) u = - 14 u

Simplify

uu - \frac{1}{u} u - 5 (u + \frac{1}{u}) u = - 14 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 - 5 (u + \frac{1}{u}) u = - 14 u

u^{2} - 1 - 5 (u + \frac{1}{u}) u = - 14 u

u^{2} - 1 - 5 (u + \frac{1}{u}) u = - 14 u

Expand

u^{2} - 1 - 5 (u + \frac{1}{u}) u : - 4 u^{2} - 6

u^{2} - 1 - 5 (u + \frac{1}{u}) u

= u^{2} - 1 - 5 u (u + \frac{1}{u})

Expand

- 5 u (u + \frac{1}{u}) : - 5 u^{2} - 5

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

Apply the distributive law:

a (b + c) = ab + a c

a = - 5 u, b = u, c = \frac{1}{u}

= - 5 uu + (- 5 u) \frac{1}{u}

Apply minus-plus rules

+ (- a) = - a

= - 5 uu - 5 \cdot \frac{1}{u} u

Simplify

- 5 uu - 5 \cdot \frac{1}{u} u : - 5 u^{2} - 5

- 5 uu - 5 \cdot \frac{1}{u} u

5 uu = 5 u^{2}

5 uu

Apply exponent rule:

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

uu = u^{1 + 1}

= 5 u^{1 + 1}

Add the numbers:

1 + 1 = 2

= 5 u^{2}

5 \cdot \frac{1}{u} u = 5

5 \cdot \frac{1}{u} u

Multiply fractions:

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

= \frac{1 \cdot 5 u}{u}

Cancel the common factor:

u

= 1 \cdot 5

Multiply the numbers:

1 \cdot 5 = 5

= 5

= - 5 u^{2} - 5

= - 5 u^{2} - 5

= u^{2} - 1 - 5 u^{2} - 5

Simplify

u^{2} - 1 - 5 u^{2} - 5 : - 4 u^{2} - 6

u^{2} - 1 - 5 u^{2} - 5

Group like terms

= u^{2} - 5 u^{2} - 1 - 5

Add similar elements:

u^{2} - 5 u^{2} = - 4 u^{2}

= - 4 u^{2} - 1 - 5

Subtract the numbers:

- 1 - 5 = - 6

= - 4 u^{2} - 6

= - 4 u^{2} - 6

- 4 u^{2} - 6 = - 14 u

Solve

- 4 u^{2} - 6 = - 14 u : u = \frac{1}{2}, u = 3

- 4 u^{2} - 6 = - 14 u

Move

14 u

to the left side

- 4 u^{2} - 6 = - 14 u

Add

14 u

to both sides

- 4 u^{2} - 6 + 14 u = - 14 u + 14 u

Simplify

- 4 u^{2} - 6 + 14 u = 0

- 4 u^{2} - 6 + 14 u = 0

Write in the standard form

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

- 4 u^{2} + 14 u - 6 = 0

Solve with the quadratic formula

- 4 u^{2} + 14 u - 6 = 0

Quadratic Equation Formula:

For

a = - 4, b = 14, c = - 6

u_{1, 2} = \frac{- 14 \pm 1 4 ^{2} - 4 ( - 4 ) ( - 6 )}{2 ( - 4 )}

u_{1, 2} = \frac{- 14 \pm 1 4 ^{2} - 4 ( - 4 ) ( - 6 )}{2 ( - 4 )}

1 4^{2} - 4 (- 4) (- 6) = 10

1 4^{2} - 4 (- 4) (- 6)

Apply rule

- (- a) = a

= 1 4^{2} - 4 \cdot 4 \cdot 6

Multiply the numbers:

4 \cdot 4 \cdot 6 = 96

= 1 4^{2} - 96

1 4^{2} = 196

= 196 - 96

Subtract the numbers:

196 - 96 = 100

= 100

Factor the number:

100 = 1 0^{2}

= 1 0^{2}

Apply radical rule:

1 0^{2} = 10

= 10

u_{1, 2} = \frac{- 14 \pm 10}{2 ( - 4 )}

Separate the solutions

u_{1} = \frac{- 14 + 10}{2 ( - 4 )}, u_{2} = \frac{- 14 - 10}{2 ( - 4 )}

u = \frac{- 14 + 10}{2 ( - 4 )} : \frac{1}{2}

\frac{- 14 + 10}{2 ( - 4 )}

Remove parentheses:

(- a) = - a

= \frac{- 14 + 10}{- 2 \cdot 4}

Add/Subtract the numbers:

- 14 + 10 = - 4

= \frac{- 4}{- 2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{- 4}{- 8}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{4}{8}

Cancel the common factor:

4

= \frac{1}{2}

u = \frac{- 14 - 10}{2 ( - 4 )} : 3

\frac{- 14 - 10}{2 ( - 4 )}

Remove parentheses:

(- a) = - a

= \frac{- 14 - 10}{- 2 \cdot 4}

Subtract the numbers:

- 14 - 10 = - 24

= \frac{- 24}{- 2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{- 24}{- 8}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{24}{8}

Divide the numbers:

\frac{24}{8} = 3

= 3

The solutions to the quadratic equation are:

u = \frac{1}{2}, u = 3

u = \frac{1}{2}, u = 3

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

u - u^{- 1} - 5 (u + u^{- 1})

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = \frac{1}{2}, u = 3

u = \frac{1}{2}, u = 3

Substitute back

u = e^{x},

solve for

x

Solve

e^{x} = \frac{1}{2} : x = - ln (2)

e^{x} = \frac{1}{2}

Apply exponent rules

e^{x} = \frac{1}{2}

f (x) = g (x)

, then

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

ln (e^{x}) = ln (\frac{1}{2})

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (\frac{1}{2})

Simplify

ln (\frac{1}{2}) : - ln (2)

ln (\frac{1}{2})

Apply log rule:

lo g_{a} (\frac{1}{x}) = - lo g_{a} (x)

= - ln (2)

x = - ln (2)

x = - ln (2)

Solve

e^{x} = 3 : x = ln (3)

e^{x} = 3

Apply exponent rules

e^{x} = 3

f (x) = g (x)

, then

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

ln (e^{x}) = ln (3)

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (3)

x = ln (3)

x = - ln (2), x = ln (3)

x = - ln (2), x = ln (3)

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Frequently Asked Questions (FAQ)

What is the general solution for sinh(x)-5cosh(x)=-7 ?
The general solution for sinh(x)-5cosh(x)=-7 is x=-ln(2),x=ln(3)