What is the general solution for 9sinh(x)-5cosh(x)=15 ?

The general solution for 9sinh(x)-5cosh(x)=15 is x=ln((15+sqrt(281))/4)

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9sinh(x)-5cosh(x)=15

Solution

9 sinh (x) - 5 cosh (x) = 15

Solution

x = ln (\frac{15 + 281}{4})

Degrees

x = 118.71739 \dots^{\circ}

Solution steps

9 sinh (x) - 5 cosh (x) = 15

Rewrite using trig identities

9 sinh (x) - 5 cosh (x) = 15

Use the Hyperbolic identity:

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

9 \cdot \frac{e ^{x} - e ^{- x}}{2} - 5 cosh (x) = 15

Use the Hyperbolic identity:

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

9 \cdot \frac{e ^{x} - e ^{- x}}{2} - 5 \cdot \frac{e ^{x} + e ^{- x}}{2} = 15

9 \cdot \frac{e ^{x} - e ^{- x}}{2} - 5 \cdot \frac{e ^{x} + e ^{- x}}{2} = 15

9 \cdot \frac{e ^{x} - e ^{- x}}{2} - 5 \cdot \frac{e ^{x} + e ^{- x}}{2} = 15 : x = ln (\frac{15 + 281}{4})

9 \cdot \frac{e ^{x} - e ^{- x}}{2} - 5 \cdot \frac{e ^{x} + e ^{- x}}{2} = 15

Apply exponent rules

9 \cdot \frac{e ^{x} - e ^{- x}}{2} - 5 \cdot \frac{e ^{x} + e ^{- x}}{2} = 15

Apply exponent rule:

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

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

9 \cdot \frac{e ^{x} - ( e ^{x} ) ^{- 1}}{2} - 5 \cdot \frac{e ^{x} + ( e ^{x} ) ^{- 1}}{2} = 15

9 \cdot \frac{e ^{x} - ( e ^{x} ) ^{- 1}}{2} - 5 \cdot \frac{e ^{x} + ( e ^{x} ) ^{- 1}}{2} = 15

Rewrite the equation with

e^{x} = u

9 \cdot \frac{u - ( u ) ^{- 1}}{2} - 5 \cdot \frac{u + ( u ) ^{- 1}}{2} = 15

Solve

9 \cdot \frac{u - u ^{- 1}}{2} - 5 \cdot \frac{u + u ^{- 1}}{2} = 15 : u = \frac{15 + 281}{4}, u = \frac{15 - 281}{4}

9 \cdot \frac{u - u ^{- 1}}{2} - 5 \cdot \frac{u + u ^{- 1}}{2} = 15

Refine

\frac{9 ( u ^{2} - 1 )}{2 u} - \frac{5 ( u ^{2} + 1 )}{2 u} = 15

Multiply both sides by

2 u

\frac{9 ( u ^{2} - 1 )}{2 u} - \frac{5 ( u ^{2} + 1 )}{2 u} = 15

Multiply both sides by

2 u

\frac{9 ( u ^{2} - 1 )}{2 u} \cdot 2 u - \frac{5 ( u ^{2} + 1 )}{2 u} \cdot 2 u = 15 \cdot 2 u

Simplify

\frac{9 ( u ^{2} - 1 )}{2 u} \cdot 2 u - \frac{5 ( u ^{2} + 1 )}{2 u} \cdot 2 u = 15 \cdot 2 u

Simplify

\frac{9 ( u ^{2} - 1 )}{2 u} \cdot 2 u : 9 (u^{2} - 1)

\frac{9 ( u ^{2} - 1 )}{2 u} \cdot 2 u

Multiply fractions:

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

= \frac{9 ( u ^{2} - 1 ) \cdot 2 u}{2 u}

Cancel the common factor:

2

= \frac{9 ( u ^{2} - 1 ) u}{u}

Cancel the common factor:

u

= 9 (u^{2} - 1)

Simplify

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

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

Multiply fractions:

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

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

Cancel the common factor:

2

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

Cancel the common factor:

u

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

Simplify

15 \cdot 2 u : 30 u

15 \cdot 2 u

Multiply the numbers:

15 \cdot 2 = 30

= 30 u

9 (u^{2} - 1) - 5 (u^{2} + 1) = 30 u

9 (u^{2} - 1) - 5 (u^{2} + 1) = 30 u

9 (u^{2} - 1) - 5 (u^{2} + 1) = 30 u

Solve

9 (u^{2} - 1) - 5 (u^{2} + 1) = 30 u : u = \frac{15 + 281}{4}, u = \frac{15 - 281}{4}

9 (u^{2} - 1) - 5 (u^{2} + 1) = 30 u

Expand

9 (u^{2} - 1) - 5 (u^{2} + 1) : 4 u^{2} - 14

9 (u^{2} - 1) - 5 (u^{2} + 1)

Expand

9 (u^{2} - 1) : 9 u^{2} - 9

9 (u^{2} - 1)

Apply the distributive law:

a (b - c) = ab - a c

a = 9, b = u^{2}, c = 1

= 9 u^{2} - 9 \cdot 1

Multiply the numbers:

9 \cdot 1 = 9

= 9 u^{2} - 9

= 9 u^{2} - 9 - 5 (u^{2} + 1)

Expand

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

- 5 (u^{2} + 1)

Apply the distributive law:

a (b + c) = ab + a c

a = - 5, b = u^{2}, c = 1

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

Apply minus-plus rules

+ (- a) = - a

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

Multiply the numbers:

5 \cdot 1 = 5

= - 5 u^{2} - 5

= 9 u^{2} - 9 - 5 u^{2} - 5

Simplify

9 u^{2} - 9 - 5 u^{2} - 5 : 4 u^{2} - 14

9 u^{2} - 9 - 5 u^{2} - 5

Group like terms

= 9 u^{2} - 5 u^{2} - 9 - 5

Add similar elements:

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

= 4 u^{2} - 9 - 5

Subtract the numbers:

- 9 - 5 = - 14

= 4 u^{2} - 14

= 4 u^{2} - 14

4 u^{2} - 14 = 30 u

Move

30 u

to the left side

4 u^{2} - 14 = 30 u

Subtract

30 u

from both sides

4 u^{2} - 14 - 30 u = 30 u - 30 u

Simplify

4 u^{2} - 14 - 30 u = 0

4 u^{2} - 14 - 30 u = 0

Write in the standard form

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

4 u^{2} - 30 u - 14 = 0

Solve with the quadratic formula

4 u^{2} - 30 u - 14 = 0

Quadratic Equation Formula:

For

a = 4, b = - 30, c = - 14

u_{1, 2} = \frac{- ( - 30 ) \pm ( - 30 ) ^{2} - 4 \cdot 4 ( - 14 )}{2 \cdot 4}

u_{1, 2} = \frac{- ( - 30 ) \pm ( - 30 ) ^{2} - 4 \cdot 4 ( - 14 )}{2 \cdot 4}

(- 30)^{2} - 4 \cdot 4 (- 14) = 2281

(- 30)^{2} - 4 \cdot 4 (- 14)

Apply rule

- (- a) = a

= (- 30)^{2} + 4 \cdot 4 \cdot 14

Apply exponent rule:

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

n

is even

(- 30)^{2} = 3 0^{2}

= 3 0^{2} + 4 \cdot 4 \cdot 14

Multiply the numbers:

4 \cdot 4 \cdot 14 = 224

= 3 0^{2} + 224

3 0^{2} = 900

= 900 + 224

Add the numbers:

900 + 224 = 1124

= 1124

Prime factorization of

1124 : 2^{2} \cdot 281

1124

1124

divides by

21124 = 562 \cdot 2

= 2 \cdot 562

562

divides by

2562 = 281 \cdot 2

= 2 \cdot 2 \cdot 281

2, 281

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 281

= 2^{2} \cdot 281

= 2^{2} \cdot 281

Apply radical rule:

= 281 2^{2}

Apply radical rule:

2^{2} = 2

= 2281

u_{1, 2} = \frac{- ( - 30 ) \pm 2 281}{2 \cdot 4}

Separate the solutions

u_{1} = \frac{- ( - 30 ) + 2 281}{2 \cdot 4}, u_{2} = \frac{- ( - 30 ) - 2 281}{2 \cdot 4}

u = \frac{- ( - 30 ) + 2 281}{2 \cdot 4} : \frac{15 + 281}{4}

\frac{- ( - 30 ) + 2 281}{2 \cdot 4}

Apply rule

- (- a) = a

= \frac{30 + 2 281}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{30 + 2 281}{8}

Factor

30 + 2281 : 2 (15 + 281)

30 + 2281

Rewrite as

= 2 \cdot 15 + 2281

Factor out common term

2

= 2 (15 + 281)

= \frac{2 ( 15 + 281 )}{8}

Cancel the common factor:

2

= \frac{15 + 281}{4}

u = \frac{- ( - 30 ) - 2 281}{2 \cdot 4} : \frac{15 - 281}{4}

\frac{- ( - 30 ) - 2 281}{2 \cdot 4}

Apply rule

- (- a) = a

= \frac{30 - 2 281}{2 \cdot 4}

Multiply the numbers:

2 \cdot 4 = 8

= \frac{30 - 2 281}{8}

Factor

30 - 2281 : 2 (15 - 281)

30 - 2281

Rewrite as

= 2 \cdot 15 - 2281

Factor out common term

2

= 2 (15 - 281)

= \frac{2 ( 15 - 281 )}{8}

Cancel the common factor:

2

= \frac{15 - 281}{4}

The solutions to the quadratic equation are:

u = \frac{15 + 281}{4}, u = \frac{15 - 281}{4}

u = \frac{15 + 281}{4}, u = \frac{15 - 281}{4}

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

9 \frac{u - u ^{- 1}}{2} - 5 \frac{u + u ^{- 1}}{2}

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = \frac{15 + 281}{4}, u = \frac{15 - 281}{4}

u = \frac{15 + 281}{4}, u = \frac{15 - 281}{4}

Substitute back

u = e^{x},

solve for

x

Solve

e^{x} = \frac{15 + 281}{4} : x = ln (\frac{15 + 281}{4})

e^{x} = \frac{15 + 281}{4}

Apply exponent rules

e^{x} = \frac{15 + 281}{4}

f (x) = g (x)

, then

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

ln (e^{x}) = ln (\frac{15 + 281}{4})

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (\frac{15 + 281}{4})

x = ln (\frac{15 + 281}{4})

Solve

e^{x} = \frac{15 - 281}{4} :

No Solution for

x \in R

e^{x} = \frac{15 - 281}{4}

a^{f (x)}

cannot be zero or negative for

x \in R

No Solution for x \in R

x = ln (\frac{15 + 281}{4})

x = ln (\frac{15 + 281}{4})

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Frequently Asked Questions (FAQ)

What is the general solution for 9sinh(x)-5cosh(x)=15 ?
The general solution for 9sinh(x)-5cosh(x)=15 is x=ln((15+sqrt(281))/4)