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cosh^2(x)-10cosh(x)+25=0

Solution

cosh^{2} (x) - 10 cosh (x) + 25 = 0

Solution

x = ln (0.10101 \dots), x = ln (0.10102 \dots), x = ln (9.89897 \dots), x = ln (\frac{9701}{980})

Degrees

x = - 131.34710 \dots^{\circ}, x = - 131.34621 \dots^{\circ}, x = 131.34665 \dots^{\circ}, x = 131.34666 \dots^{\circ}

Solution steps

cosh^{2} (x) - 10 cosh (x) + 25 = 0

Rewrite using trig identities

cosh^{2} (x) - 10 cosh (x) + 25 = 0

Use the Hyperbolic identity:

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

(\frac{e ^{x} + e ^{- x}}{2})^{2} - 10 \cdot \frac{e ^{x} + e ^{- x}}{2} + 25 = 0

(\frac{e ^{x} + e ^{- x}}{2})^{2} - 10 \cdot \frac{e ^{x} + e ^{- x}}{2} + 25 = 0

(\frac{e ^{x} + e ^{- x}}{2})^{2} - 10 \cdot \frac{e ^{x} + e ^{- x}}{2} + 25 = 0 : x = ln (0.10101 \dots), x = ln (0.10102 \dots), x = ln (9.89897 \dots), x = ln (\frac{9701}{980})

(\frac{e ^{x} + e ^{- x}}{2})^{2} - 10 \cdot \frac{e ^{x} + e ^{- x}}{2} + 25 = 0

Apply exponent rules

(\frac{e ^{x} + e ^{- x}}{2})^{2} - 10 \cdot \frac{e ^{x} + e ^{- x}}{2} + 25 = 0

Apply exponent rule:

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

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

(\frac{e ^{x} + ( e ^{x} ) ^{- 1}}{2})^{2} - 10 \cdot \frac{e ^{x} + ( e ^{x} ) ^{- 1}}{2} + 25 = 0

(\frac{e ^{x} + ( e ^{x} ) ^{- 1}}{2})^{2} - 10 \cdot \frac{e ^{x} + ( e ^{x} ) ^{- 1}}{2} + 25 = 0

Rewrite the equation with

e^{x} = u

(\frac{u + ( u ) ^{- 1}}{2})^{2} - 10 \cdot \frac{u + ( u ) ^{- 1}}{2} + 25 = 0

Solve

(\frac{u + u ^{- 1}}{2})^{2} - 10 \cdot \frac{u + u ^{- 1}}{2} + 25 = 0 : u \approx 0.10101 \dots, u \approx 0.10102 \dots, u \approx 9.89897 \dots, u = \frac{9701}{980}

(\frac{u + u ^{- 1}}{2})^{2} - 10 \cdot \frac{u + u ^{- 1}}{2} + 25 = 0

Refine

\frac{( u ^{2} + 1 ) ^{2}}{4 u ^{2}} - \frac{5 ( u ^{2} + 1 )}{u} + 25 = 0

Multiply by LCM

\frac{( u ^{2} + 1 ) ^{2}}{4 u ^{2}} - \frac{5 ( u ^{2} + 1 )}{u} + 25 = 0

Find Least Common Multiplier of

4 u^{2}, u : 4 u^{2}

4 u^{2}, u

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

4 u^{2}

u

= 4 u^{2}

Multiply by LCM=

4 u^{2}

\frac{( u ^{2} + 1 ) ^{2}}{4 u ^{2}} \cdot 4 u^{2} - \frac{5 ( u ^{2} + 1 )}{u} \cdot 4 u^{2} + 25 \cdot 4 u^{2} = 0 \cdot 4 u^{2}

Simplify

\frac{( u ^{2} + 1 ) ^{2}}{4 u ^{2}} \cdot 4 u^{2} - \frac{5 ( u ^{2} + 1 )}{u} \cdot 4 u^{2} + 25 \cdot 4 u^{2} = 0 \cdot 4 u^{2}

Simplify

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

\frac{( u ^{2} + 1 ) ^{2}}{4 u ^{2}} \cdot 4 u^{2}

Multiply fractions:

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

= \frac{( u ^{2} + 1 ) ^{2} \cdot 4 u ^{2}}{4 u ^{2}}

Cancel the common factor:

4

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

Cancel the common factor:

u^{2}

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

Simplify

- \frac{5 ( u ^{2} + 1 )}{u} \cdot 4 u^{2} : - 20 u (u^{2} + 1)

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

Multiply fractions:

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

= - \frac{5 ( u ^{2} + 1 ) \cdot 4 u ^{2}}{u}

Multiply the numbers:

5 \cdot 4 = 20

= - \frac{20 u ^{2} ( u ^{2} + 1 )}{u}

Cancel the common factor:

u

= - 20 u (u^{2} + 1)

Simplify

25 \cdot 4 u^{2} : 100 u^{2}

25 \cdot 4 u^{2}

Multiply the numbers:

25 \cdot 4 = 100

= 100 u^{2}

Simplify

0 \cdot 4 u^{2} : 0

0 \cdot 4 u^{2}

Apply rule

0 \cdot a = 0

= 0

(u^{2} + 1)^{2} - 20 u (u^{2} + 1) + 100 u^{2} = 0

(u^{2} + 1)^{2} - 20 u (u^{2} + 1) + 100 u^{2} = 0

(u^{2} + 1)^{2} - 20 u (u^{2} + 1) + 100 u^{2} = 0

Solve

(u^{2} + 1)^{2} - 20 u (u^{2} + 1) + 100 u^{2} = 0 : u \approx 0.10101 \dots, u \approx 0.10102 \dots, u \approx 9.89897 \dots, u = \frac{9701}{980}

(u^{2} + 1)^{2} - 20 u (u^{2} + 1) + 100 u^{2} = 0

Expand

(u^{2} + 1)^{2} - 20 u (u^{2} + 1) + 100 u^{2} : u^{4} - 20 u^{3} + 102 u^{2} - 20 u + 1

(u^{2} + 1)^{2} - 20 u (u^{2} + 1) + 100 u^{2}

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

Apply Perfect Square Formula:

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

a = u^{2}, b = 1

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

Simplify

(u^{2})^{2} + 2 u^{2} \cdot 1 + 1^{2} : u^{4} + 2 u^{2} + 1

(u^{2})^{2} + 2 u^{2} \cdot 1 + 1^{2}

Apply rule

1^{a} = 1

1^{2} = 1

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

(u^{2})^{2} = u^{4}

(u^{2})^{2}

Apply exponent rule:

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

= u^{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= u^{4}

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

2 u^{2} \cdot 1

Multiply the numbers:

2 \cdot 1 = 2

= 2 u^{2}

= u^{4} + 2 u^{2} + 1

= u^{4} + 2 u^{2} + 1

= u^{4} + 2 u^{2} + 1 - 20 u (u^{2} + 1) + 100 u^{2}

Expand

- 20 u (u^{2} + 1) : - 20 u^{3} - 20 u

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

Apply the distributive law:

a (b + c) = ab + a c

a = - 20 u, b = u^{2}, c = 1

= - 20 u u^{2} + (- 20 u) \cdot 1

Apply minus-plus rules

+ (- a) = - a

= - 20 u^{2} u - 20 \cdot 1 \cdot u

Simplify

- 20 u^{2} u - 20 \cdot 1 \cdot u : - 20 u^{3} - 20 u

- 20 u^{2} u - 20 \cdot 1 \cdot u

20 u^{2} u = 20 u^{3}

20 u^{2} u

Apply exponent rule:

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

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

= 20 u^{2 + 1}

Add the numbers:

2 + 1 = 3

= 20 u^{3}

20 \cdot 1 \cdot u = 20 u

20 \cdot 1 \cdot u

Multiply the numbers:

20 \cdot 1 = 20

= 20 u

= - 20 u^{3} - 20 u

= - 20 u^{3} - 20 u

= u^{4} + 2 u^{2} + 1 - 20 u^{3} - 20 u + 100 u^{2}

Simplify

u^{4} + 2 u^{2} + 1 - 20 u^{3} - 20 u + 100 u^{2} : u^{4} - 20 u^{3} + 102 u^{2} - 20 u + 1

u^{4} + 2 u^{2} + 1 - 20 u^{3} - 20 u + 100 u^{2}

Group like terms

= u^{4} - 20 u^{3} + 2 u^{2} + 100 u^{2} - 20 u + 1

Add similar elements:

2 u^{2} + 100 u^{2} = 102 u^{2}

= u^{4} - 20 u^{3} + 102 u^{2} - 20 u + 1

= u^{4} - 20 u^{3} + 102 u^{2} - 20 u + 1

u^{4} - 20 u^{3} + 102 u^{2} - 20 u + 1 = 0

Find one solution for

u^{4} - 20 u^{3} + 102 u^{2} - 20 u + 1 = 0

using Newton-Raphson:

u \approx 0.10101 \dots

u^{4} - 20 u^{3} + 102 u^{2} - 20 u + 1 = 0

Newton-Raphson Approximation Definition

f (u) = u^{4} - 20 u^{3} + 102 u^{2} - 20 u + 1

Find

f^{^{'}} (u) : 4 u^{3} - 60 u^{2} + 204 u - 20

\frac{d}{d u} (u^{4} - 20 u^{3} + 102 u^{2} - 20 u + 1)

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d u} (u^{4}) - \frac{d}{d u} (20 u^{3}) + \frac{d}{d u} (102 u^{2}) - \frac{d}{d u} (20 u) + \frac{d}{d u} (1)

\frac{d}{d u} (u^{4}) = 4 u^{3}

\frac{d}{d u} (u^{4})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 4 u^{4 - 1}

Simplify

= 4 u^{3}

\frac{d}{d u} (20 u^{3}) = 60 u^{2}

\frac{d}{d u} (20 u^{3})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 20 \frac{d}{d u} (u^{3})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 20 \cdot 3 u^{3 - 1}

Simplify

= 60 u^{2}

\frac{d}{d u} (102 u^{2}) = 204 u

\frac{d}{d u} (102 u^{2})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 102 \frac{d}{d u} (u^{2})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 102 \cdot 2 u^{2 - 1}

Simplify

= 204 u

\frac{d}{d u} (20 u) = 20

\frac{d}{d u} (20 u)

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 20 \frac{d u}{d u}

Apply the common derivative:

\frac{d u}{d u} = 1

= 20 \cdot 1

Simplify

= 20

\frac{d}{d u} (1) = 0

\frac{d}{d u} (1)

Derivative of a constant:

\frac{d}{d x} (a) = 0

= 0

= 4 u^{3} - 60 u^{2} + 204 u - 20 + 0

Simplify

= 4 u^{3} - 60 u^{2} + 204 u - 20

Let

u_{0} = 0

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = 0.05 : Δ u_{1} = 0.05

f (u_{0}) = 0^{4} - 20 \cdot 0^{3} + 102 \cdot 0^{2} - 20 \cdot 0 + 1 = 1

f^{^{'}} (u_{0}) = 4 \cdot 0^{3} - 60 \cdot 0^{2} + 204 \cdot 0 - 20 = - 20

u_{1} = 0.05

Δ u_{1} = ∣ 0.05 - 0 ∣ = 0.05

Δ u_{1} = 0.05

u_{2} = 0.07537 \dots : Δ u_{2} = 0.02537 \dots

f (u_{1}) = 0.0 5^{4} - 20 \cdot 0.0 5^{3} + 102 \cdot 0.0 5^{2} - 20 \cdot 0.05 + 1 = 0.25250625

f^{^{'}} (u_{1}) = 4 \cdot 0.0 5^{3} - 60 \cdot 0.0 5^{2} + 204 \cdot 0.05 - 20 = - 9.9495

u_{2} = 0.07537 \dots

Δ u_{2} = ∣ 0.07537 \dots - 0.05 ∣ = 0.02537 \dots

Δ u_{2} = 0.02537 \dots

u_{3} = 0.08816 \dots : Δ u_{3} = 0.01278 \dots

f (u_{2}) = 0.07537 \dots^{4} - 20 \cdot 0.07537 \dots^{3} + 102 \cdot 0.07537 \dots^{2} - 20 \cdot 0.07537 \dots + 1 = 0.06345 \dots

f^{^{'}} (u_{2}) = 4 \cdot 0.07537 \dots^{3} - 60 \cdot 0.07537 \dots^{2} + 204 \cdot 0.07537 \dots - 20 = - 4.96193 \dots

u_{3} = 0.08816 \dots

Δ u_{3} = ∣ 0.08816 \dots - 0.07537 \dots ∣ = 0.01278 \dots

Δ u_{3} = 0.01278 \dots

u_{4} = 0.09458 \dots : Δ u_{4} = 0.00641 \dots

f (u_{3}) = 0.08816 \dots^{4} - 20 \cdot 0.08816 \dots^{3} + 102 \cdot 0.08816 \dots^{2} - 20 \cdot 0.08816 \dots + 1 = 0.01590 \dots

f^{^{'}} (u_{3}) = 4 \cdot 0.08816 \dots^{3} - 60 \cdot 0.08816 \dots^{2} + 204 \cdot 0.08816 \dots - 20 = - 2.47773 \dots

u_{4} = 0.09458 \dots

Δ u_{4} = ∣ 0.09458 \dots - 0.08816 \dots ∣ = 0.00641 \dots

Δ u_{4} = 0.00641 \dots

u_{5} = 0.09780 \dots : Δ u_{5} = 0.00321 \dots

f (u_{4}) = 0.09458 \dots^{4} - 20 \cdot 0.09458 \dots^{3} + 102 \cdot 0.09458 \dots^{2} - 20 \cdot 0.09458 \dots + 1 = 0.00398 \dots

f^{^{'}} (u_{4}) = 4 \cdot 0.09458 \dots^{3} - 60 \cdot 0.09458 \dots^{2} + 204 \cdot 0.09458 \dots - 20 = - 1.23805 \dots

u_{5} = 0.09780 \dots

Δ u_{5} = ∣ 0.09780 \dots - 0.09458 \dots ∣ = 0.00321 \dots

Δ u_{5} = 0.00321 \dots

u_{6} = 0.09941 \dots : Δ u_{6} = 0.00160 \dots

f (u_{5}) = 0.09780 \dots^{4} - 20 \cdot 0.09780 \dots^{3} + 102 \cdot 0.09780 \dots^{2} - 20 \cdot 0.09780 \dots + 1 = 0.00099 \dots

f^{^{'}} (u_{5}) = 4 \cdot 0.09780 \dots^{3} - 60 \cdot 0.09780 \dots^{2} + 204 \cdot 0.09780 \dots - 20 = - 0.61882 \dots

u_{6} = 0.09941 \dots

Δ u_{6} = ∣ 0.09941 \dots - 0.09780 \dots ∣ = 0.00160 \dots

Δ u_{6} = 0.00160 \dots

u_{7} = 0.10021 \dots : Δ u_{7} = 0.00080 \dots

f (u_{6}) = 0.09941 \dots^{4} - 20 \cdot 0.09941 \dots^{3} + 102 \cdot 0.09941 \dots^{2} - 20 \cdot 0.09941 \dots + 1 = 0.00024 \dots

f^{^{'}} (u_{6}) = 4 \cdot 0.09941 \dots^{3} - 60 \cdot 0.09941 \dots^{2} + 204 \cdot 0.09941 \dots - 20 = - 0.30936 \dots

u_{7} = 0.10021 \dots

Δ u_{7} = ∣ 0.10021 \dots - 0.09941 \dots ∣ = 0.00080 \dots

Δ u_{7} = 0.00080 \dots

u_{8} = 0.10061 \dots : Δ u_{8} = 0.00040 \dots

f (u_{7}) = 0.10021 \dots^{4} - 20 \cdot 0.10021 \dots^{3} + 102 \cdot 0.10021 \dots^{2} - 20 \cdot 0.10021 \dots + 1 = 0.00006 \dots

f^{^{'}} (u_{7}) = 4 \cdot 0.10021 \dots^{3} - 60 \cdot 0.10021 \dots^{2} + 204 \cdot 0.10021 \dots - 20 = - 0.15466 \dots

u_{8} = 0.10061 \dots

Δ u_{8} = ∣ 0.10061 \dots - 0.10021 \dots ∣ = 0.00040 \dots

Δ u_{8} = 0.00040 \dots

u_{9} = 0.10081 \dots : Δ u_{9} = 0.00020 \dots

f (u_{8}) = 0.10061 \dots^{4} - 20 \cdot 0.10061 \dots^{3} + 102 \cdot 0.10061 \dots^{2} - 20 \cdot 0.10061 \dots + 1 = 0.00001 \dots

f^{^{'}} (u_{8}) = 4 \cdot 0.10061 \dots^{3} - 60 \cdot 0.10061 \dots^{2} + 204 \cdot 0.10061 \dots - 20 = - 0.07733 \dots

u_{9} = 0.10081 \dots

Δ u_{9} = ∣ 0.10081 \dots - 0.10061 \dots ∣ = 0.00020 \dots

Δ u_{9} = 0.00020 \dots

u_{10} = 0.10091 \dots : Δ u_{10} = 0.00010 \dots

f (u_{9}) = 0.10081 \dots^{4} - 20 \cdot 0.10081 \dots^{3} + 102 \cdot 0.10081 \dots^{2} - 20 \cdot 0.10081 \dots + 1 = 3.8928 E - 6

f^{^{'}} (u_{9}) = 4 \cdot 0.10081 \dots^{3} - 60 \cdot 0.10081 \dots^{2} + 204 \cdot 0.10081 \dots - 20 = - 0.03866 \dots

u_{10} = 0.10091 \dots

Δ u_{10} = ∣ 0.10091 \dots - 0.10081 \dots ∣ = 0.00010 \dots

Δ u_{10} = 0.00010 \dots

u_{11} = 0.10097 \dots : Δ u_{11} = 0.00005 \dots

f (u_{10}) = 0.10091 \dots^{4} - 20 \cdot 0.10091 \dots^{3} + 102 \cdot 0.10091 \dots^{2} - 20 \cdot 0.10091 \dots + 1 = 9.73221 E - 7

f^{^{'}} (u_{10}) = 4 \cdot 0.10091 \dots^{3} - 60 \cdot 0.10091 \dots^{2} + 204 \cdot 0.10091 \dots - 20 = - 0.01933 \dots

u_{11} = 0.10097 \dots

Δ u_{11} = ∣ 0.10097 \dots - 0.10091 \dots ∣ = 0.00005 \dots

Δ u_{11} = 0.00005 \dots

u_{12} = 0.10099 \dots : Δ u_{12} = 0.00002 \dots

f (u_{11}) = 0.10097 \dots^{4} - 20 \cdot 0.10097 \dots^{3} + 102 \cdot 0.10097 \dots^{2} - 20 \cdot 0.10097 \dots + 1 = 2.43308 E - 7

f^{^{'}} (u_{11}) = 4 \cdot 0.10097 \dots^{3} - 60 \cdot 0.10097 \dots^{2} + 204 \cdot 0.10097 \dots - 20 = - 0.00966 \dots

u_{12} = 0.10099 \dots

Δ u_{12} = ∣ 0.10099 \dots - 0.10097 \dots ∣ = 0.00002 \dots

Δ u_{12} = 0.00002 \dots

u_{13} = 0.10100 \dots : Δ u_{13} = 0.00001 \dots

f (u_{12}) = 0.10099 \dots^{4} - 20 \cdot 0.10099 \dots^{3} + 102 \cdot 0.10099 \dots^{2} - 20 \cdot 0.10099 \dots + 1 = 6.08272 E - 8

f^{^{'}} (u_{12}) = 4 \cdot 0.10099 \dots^{3} - 60 \cdot 0.10099 \dots^{2} + 204 \cdot 0.10099 \dots - 20 = - 0.00483 \dots

u_{13} = 0.10100 \dots

Δ u_{13} = ∣ 0.10100 \dots - 0.10099 \dots ∣ = 0.00001 \dots

Δ u_{13} = 0.00001 \dots

u_{14} = 0.10101 \dots : Δ u_{14} = 6.29293 E - 6

f (u_{13}) = 0.10100 \dots^{4} - 20 \cdot 0.10100 \dots^{3} + 102 \cdot 0.10100 \dots^{2} - 20 \cdot 0.10100 \dots + 1 = 1.52068 E - 8

f^{^{'}} (u_{13}) = 4 \cdot 0.10100 \dots^{3} - 60 \cdot 0.10100 \dots^{2} + 204 \cdot 0.10100 \dots - 20 = - 0.00241 \dots

u_{14} = 0.10101 \dots

Δ u_{14} = ∣ 0.10101 \dots - 0.10100 \dots ∣ = 6.29293 E - 6

Δ u_{14} = 6.29293 E - 6

u_{15} = 0.10101 \dots : Δ u_{15} = 3.14647 E - 6

f (u_{14}) = 0.10101 \dots^{4} - 20 \cdot 0.10101 \dots^{3} + 102 \cdot 0.10101 \dots^{2} - 20 \cdot 0.10101 \dots + 1 = 3.80172 E - 9

f^{^{'}} (u_{14}) = 4 \cdot 0.10101 \dots^{3} - 60 \cdot 0.10101 \dots^{2} + 204 \cdot 0.10101 \dots - 20 = - 0.00120 \dots

u_{15} = 0.10101 \dots

Δ u_{15} = ∣ 0.10101 \dots - 0.10101 \dots ∣ = 3.14647 E - 6

Δ u_{15} = 3.14647 E - 6

u_{16} = 0.10101 \dots : Δ u_{16} = 1.57324 E - 6

f (u_{15}) = 0.10101 \dots^{4} - 20 \cdot 0.10101 \dots^{3} + 102 \cdot 0.10101 \dots^{2} - 20 \cdot 0.10101 \dots + 1 = 9.5043 E - 10

f^{^{'}} (u_{15}) = 4 \cdot 0.10101 \dots^{3} - 60 \cdot 0.10101 \dots^{2} + 204 \cdot 0.10101 \dots - 20 = - 0.00060 \dots

u_{16} = 0.10101 \dots

Δ u_{16} = ∣ 0.10101 \dots - 0.10101 \dots ∣ = 1.57324 E - 6

Δ u_{16} = 1.57324 E - 6

u_{17} = 0.10101 \dots : Δ u_{17} = 7.86619 E - 7

f (u_{16}) = 0.10101 \dots^{4} - 20 \cdot 0.10101 \dots^{3} + 102 \cdot 0.10101 \dots^{2} - 20 \cdot 0.10101 \dots + 1 = 2.37607 E - 10

f^{^{'}} (u_{16}) = 4 \cdot 0.10101 \dots^{3} - 60 \cdot 0.10101 \dots^{2} + 204 \cdot 0.10101 \dots - 20 = - 0.00030 \dots

u_{17} = 0.10101 \dots

Δ u_{17} = ∣ 0.10101 \dots - 0.10101 \dots ∣ = 7.86619 E - 7

Δ u_{17} = 7.86619 E - 7

u \approx 0.10101 \dots

Apply long division:

\frac{u ^{4} - 20 u ^{3} + 102 u ^{2} - 20 u + 1}{u - 0.10101 \dots} = u^{3} - 19.89898 \dots u^{2} + 99.98981 \dots u - 9.89905 \dots

u^{3} - 19.89898 \dots u^{2} + 99.98981 \dots u - 9.89905 \dots \approx 0

Find one solution for

u^{3} - 19.89898 \dots u^{2} + 99.98981 \dots u - 9.89905 \dots = 0

using Newton-Raphson:

u \approx 0.10102 \dots

u^{3} - 19.89898 \dots u^{2} + 99.98981 \dots u - 9.89905 \dots = 0

Newton-Raphson Approximation Definition

f (u) = u^{3} - 19.89898 \dots u^{2} + 99.98981 \dots u - 9.89905 \dots

Find

f^{^{'}} (u) : 3 u^{2} - 39.79796 \dots u + 99.98981 \dots

\frac{d}{d u} (u^{3} - 19.89898 \dots u^{2} + 99.98981 \dots u - 9.89905 \dots)

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d u} (u^{3}) - \frac{d}{d u} (19.89898 \dots u^{2}) + \frac{d}{d u} (99.98981 \dots u) - \frac{d}{d u} (9.89905 \dots)

\frac{d}{d u} (u^{3}) = 3 u^{2}

\frac{d}{d u} (u^{3})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 3 u^{3 - 1}

Simplify

= 3 u^{2}

\frac{d}{d u} (19.89898 \dots u^{2}) = 39.79796 \dots u

\frac{d}{d u} (19.89898 \dots u^{2})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 19.89898 \dots \frac{d}{d u} (u^{2})

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 19.89898 \dots \cdot 2 u^{2 - 1}

Simplify

= 39.79796 \dots u

\frac{d}{d u} (99.98981 \dots u) = 99.98981 \dots

\frac{d}{d u} (99.98981 \dots u)

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 99.98981 \dots \frac{d u}{d u}

Apply the common derivative:

\frac{d u}{d u} = 1

= 99.98981 \dots \cdot 1

Simplify

= 99.98981 \dots

\frac{d}{d u} (9.89905 \dots) = 0

\frac{d}{d u} (9.89905 \dots)

Derivative of a constant:

\frac{d}{d x} (a) = 0

= 0

= 3 u^{2} - 39.79796 \dots u + 99.98981 \dots - 0

Simplify

= 3 u^{2} - 39.79796 \dots u + 99.98981 \dots

Let

u_{0} = 0

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = 0.09900 \dots : Δ u_{1} = 0.09900 \dots

f (u_{0}) = 0^{3} - 19.89898 \dots \cdot 0^{2} + 99.98981 \dots \cdot 0 - 9.89905 \dots = - 9.89905 \dots

f^{^{'}} (u_{0}) = 3 \cdot 0^{2} - 39.79796 \dots \cdot 0 + 99.98981 \dots = 99.98981 \dots

u_{1} = 0.09900 \dots

Δ u_{1} = ∣ 0.09900 \dots - 0 ∣ = 0.09900 \dots

Δ u_{1} = 0.09900 \dots

u_{2} = 0.10102 \dots : Δ u_{2} = 0.00201 \dots

f (u_{1}) = 0.09900 \dots^{3} - 19.89898 \dots \cdot 0.09900 \dots^{2} + 99.98981 \dots \cdot 0.09900 \dots - 9.89905 \dots = - 0.19406 \dots

f^{^{'}} (u_{1}) = 3 \cdot 0.09900 \dots^{2} - 39.79796 \dots \cdot 0.09900 \dots + 99.98981 \dots = 96.07918 \dots

u_{2} = 0.10102 \dots

Δ u_{2} = ∣ 0.10102 \dots - 0.09900 \dots ∣ = 0.00201 \dots

Δ u_{2} = 0.00201 \dots

u_{3} = 0.10102 \dots : Δ u_{3} = 8.32927 E - 7

f (u_{2}) = 0.10102 \dots^{3} - 19.89898 \dots \cdot 0.10102 \dots^{2} + 99.98981 \dots \cdot 0.10102 \dots - 9.89905 \dots = - 0.00007 \dots

f^{^{'}} (u_{2}) = 3 \cdot 0.10102 \dots^{2} - 39.79796 \dots \cdot 0.10102 \dots + 99.98981 \dots = 96.00001 \dots

u_{3} = 0.10102 \dots

Δ u_{3} = ∣ 0.10102 \dots - 0.10102 \dots ∣ = 8.32927 E - 7

Δ u_{3} = 8.32927 E - 7

u \approx 0.10102 \dots

Apply long division:

\frac{u ^{3} - 19.89898 \dots u ^{2} + 99.98981 \dots u - 9.89905 \dots}{u - 0.10102 \dots} = u^{2} - 19.79795 \dots u + 97.98979 \dots

u^{2} - 19.79795 \dots u + 97.98979 \dots \approx 0

Find one solution for

u^{2} - 19.79795 \dots u + 97.98979 \dots = 0

using Newton-Raphson:

u \approx 9.89897 \dots

u^{2} - 19.79795 \dots u + 97.98979 \dots = 0

Newton-Raphson Approximation Definition

f (u) = u^{2} - 19.79795 \dots u + 97.98979 \dots

Find

f^{^{'}} (u) : 2 u - 19.79795 \dots

\frac{d}{d u} (u^{2} - 19.79795 \dots u + 97.98979 \dots)

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{d}{d u} (u^{2}) - \frac{d}{d u} (19.79795 \dots u) + \frac{d}{d u} (97.98979 \dots)

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

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

Apply the Power Rule:

\frac{d}{d x} (x^{a}) = a \cdot x^{a - 1}

= 2 u^{2 - 1}

Simplify

= 2 u

\frac{d}{d u} (19.79795 \dots u) = 19.79795 \dots

\frac{d}{d u} (19.79795 \dots u)

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 19.79795 \dots \frac{d u}{d u}

Apply the common derivative:

\frac{d u}{d u} = 1

= 19.79795 \dots \cdot 1

Simplify

= 19.79795 \dots

\frac{d}{d u} (97.98979 \dots) = 0

\frac{d}{d u} (97.98979 \dots)

Derivative of a constant:

\frac{d}{d x} (a) = 0

= 0

= 2 u - 19.79795 \dots + 0

Simplify

= 2 u - 19.79795 \dots

Let

u_{0} = 5

Compute

u_{n + 1}

until

Δ u_{n + 1} < 0.000001

u_{1} = 7.44948 \dots : Δ u_{1} = 2.44948 \dots

f (u_{0}) = 5^{2} - 19.79795 \dots \cdot 5 + 97.98979 \dots = 24.00000 \dots

f^{^{'}} (u_{0}) = 2 \cdot 5 - 19.79795 \dots = - 9.79795 \dots

u_{1} = 7.44948 \dots

Δ u_{1} = ∣ 7.44948 \dots - 5 ∣ = 2.44948 \dots

Δ u_{1} = 2.44948 \dots

u_{2} = 8.67423 \dots : Δ u_{2} = 1.22474 \dots

f (u_{1}) = 7.44948 \dots^{2} - 19.79795 \dots \cdot 7.44948 \dots + 97.98979 \dots = 6.00000 \dots

f^{^{'}} (u_{1}) = 2 \cdot 7.44948 \dots - 19.79795 \dots = - 4.89897 \dots

u_{2} = 8.67423 \dots

Δ u_{2} = ∣ 8.67423 \dots - 7.44948 \dots ∣ = 1.22474 \dots

Δ u_{2} = 1.22474 \dots

u_{3} = 9.28660 \dots : Δ u_{3} = 0.61237 \dots

f (u_{2}) = 8.67423 \dots^{2} - 19.79795 \dots \cdot 8.67423 \dots + 97.98979 \dots = 1.50000 \dots

f^{^{'}} (u_{2}) = 2 \cdot 8.67423 \dots - 19.79795 \dots = - 2.44948 \dots

u_{3} = 9.28660 \dots

Δ u_{3} = ∣ 9.28660 \dots - 8.67423 \dots ∣ = 0.61237 \dots

Δ u_{3} = 0.61237 \dots

u_{4} = 9.59279 \dots : Δ u_{4} = 0.30618 \dots

f (u_{3}) = 9.28660 \dots^{2} - 19.79795 \dots \cdot 9.28660 \dots + 97.98979 \dots = 0.37500 \dots

f^{^{'}} (u_{3}) = 2 \cdot 9.28660 \dots - 19.79795 \dots = - 1.22474 \dots

u_{4} = 9.59279 \dots

Δ u_{4} = ∣ 9.59279 \dots - 9.28660 \dots ∣ = 0.30618 \dots

Δ u_{4} = 0.30618 \dots

u_{5} = 9.74588 \dots : Δ u_{5} = 0.15309 \dots

f (u_{4}) = 9.59279 \dots^{2} - 19.79795 \dots \cdot 9.59279 \dots + 97.98979 \dots = 0.09375 \dots

f^{^{'}} (u_{4}) = 2 \cdot 9.59279 \dots - 19.79795 \dots = - 0.61237 \dots

u_{5} = 9.74588 \dots

Δ u_{5} = ∣ 9.74588 \dots - 9.59279 \dots ∣ = 0.15309 \dots

Δ u_{5} = 0.15309 \dots

u_{6} = 9.82243 \dots : Δ u_{6} = 0.07654 \dots

f (u_{5}) = 9.74588 \dots^{2} - 19.79795 \dots \cdot 9.74588 \dots + 97.98979 \dots = 0.02343 \dots

f^{^{'}} (u_{5}) = 2 \cdot 9.74588 \dots - 19.79795 \dots = - 0.30618 \dots

u_{6} = 9.82243 \dots

Δ u_{6} = ∣ 9.82243 \dots - 9.74588 \dots ∣ = 0.07654 \dots

Δ u_{6} = 0.07654 \dots

u_{7} = 9.86070 \dots : Δ u_{7} = 0.03827 \dots

f (u_{6}) = 9.82243 \dots^{2} - 19.79795 \dots \cdot 9.82243 \dots + 97.98979 \dots = 0.00585 \dots

f^{^{'}} (u_{6}) = 2 \cdot 9.82243 \dots - 19.79795 \dots = - 0.15309 \dots

u_{7} = 9.86070 \dots

Δ u_{7} = ∣ 9.86070 \dots - 9.82243 \dots ∣ = 0.03827 \dots

Δ u_{7} = 0.03827 \dots

u_{8} = 9.87984 \dots : Δ u_{8} = 0.01913 \dots

f (u_{7}) = 9.86070 \dots^{2} - 19.79795 \dots \cdot 9.86070 \dots + 97.98979 \dots = 0.00146 \dots

f^{^{'}} (u_{7}) = 2 \cdot 9.86070 \dots - 19.79795 \dots = - 0.07654 \dots

u_{8} = 9.87984 \dots

Δ u_{8} = ∣ 9.87984 \dots - 9.86070 \dots ∣ = 0.01913 \dots

Δ u_{8} = 0.01913 \dots

u_{9} = 9.88941 \dots : Δ u_{9} = 0.00956 \dots

f (u_{8}) = 9.87984 \dots^{2} - 19.79795 \dots \cdot 9.87984 \dots + 97.98979 \dots = 0.00036 \dots

f^{^{'}} (u_{8}) = 2 \cdot 9.87984 \dots - 19.79795 \dots = - 0.03827 \dots

u_{9} = 9.88941 \dots

Δ u_{9} = ∣ 9.88941 \dots - 9.87984 \dots ∣ = 0.00956 \dots

Δ u_{9} = 0.00956 \dots

u_{10} = 9.89419 \dots : Δ u_{10} = 0.00478 \dots

f (u_{9}) = 9.88941 \dots^{2} - 19.79795 \dots \cdot 9.88941 \dots + 97.98979 \dots = 0.00009 \dots

f^{^{'}} (u_{9}) = 2 \cdot 9.88941 \dots - 19.79795 \dots = - 0.01913 \dots

u_{10} = 9.89419 \dots

Δ u_{10} = ∣ 9.89419 \dots - 9.88941 \dots ∣ = 0.00478 \dots

Δ u_{10} = 0.00478 \dots

u_{11} = 9.89658 \dots : Δ u_{11} = 0.00239 \dots

f (u_{10}) = 9.89419 \dots^{2} - 19.79795 \dots \cdot 9.89419 \dots + 97.98979 \dots = 0.00002 \dots

f^{^{'}} (u_{10}) = 2 \cdot 9.89419 \dots - 19.79795 \dots = - 0.00956 \dots

u_{11} = 9.89658 \dots

Δ u_{11} = ∣ 9.89658 \dots - 9.89419 \dots ∣ = 0.00239 \dots

Δ u_{11} = 0.00239 \dots

u_{12} = 9.89778 \dots : Δ u_{12} = 0.00119 \dots

f (u_{11}) = 9.89658 \dots^{2} - 19.79795 \dots \cdot 9.89658 \dots + 97.98979 \dots = 5.72205 E - 6

f^{^{'}} (u_{11}) = 2 \cdot 9.89658 \dots - 19.79795 \dots = - 0.00478 \dots

u_{12} = 9.89778 \dots

Δ u_{12} = ∣ 9.89778 \dots - 9.89658 \dots ∣ = 0.00119 \dots

Δ u_{12} = 0.00119 \dots

u_{13} = 9.89838 \dots : Δ u_{13} = 0.00059 \dots

f (u_{12}) = 9.89778 \dots^{2} - 19.79795 \dots \cdot 9.89778 \dots + 97.98979 \dots = 1.43051 E - 6

f^{^{'}} (u_{12}) = 2 \cdot 9.89778 \dots - 19.79795 \dots = - 0.00239 \dots

u_{13} = 9.89838 \dots

Δ u_{13} = ∣ 9.89838 \dots - 9.89778 \dots ∣ = 0.00059 \dots

Δ u_{13} = 0.00059 \dots

u_{14} = 9.89868 \dots : Δ u_{14} = 0.00029 \dots

f (u_{13}) = 9.89838 \dots^{2} - 19.79795 \dots \cdot 9.89838 \dots + 97.98979 \dots = 3.57628 E - 7

f^{^{'}} (u_{13}) = 2 \cdot 9.89838 \dots - 19.79795 \dots = - 0.00119 \dots

u_{14} = 9.89868 \dots

Δ u_{14} = ∣ 9.89868 \dots - 9.89838 \dots ∣ = 0.00029 \dots

Δ u_{14} = 0.00029 \dots

u_{15} = 9.89882 \dots : Δ u_{15} = 0.00014 \dots

f (u_{14}) = 9.89868 \dots^{2} - 19.79795 \dots \cdot 9.89868 \dots + 97.98979 \dots = 8.94072 E - 8

f^{^{'}} (u_{14}) = 2 \cdot 9.89868 \dots - 19.79795 \dots = - 0.00059 \dots

u_{15} = 9.89882 \dots

Δ u_{15} = ∣ 9.89882 \dots - 9.89868 \dots ∣ = 0.00014 \dots

Δ u_{15} = 0.00014 \dots

u_{16} = 9.89890 \dots : Δ u_{16} = 0.00007 \dots

f (u_{15}) = 9.89882 \dots^{2} - 19.79795 \dots \cdot 9.89882 \dots + 97.98979 \dots = 2.2352 E - 8

f^{^{'}} (u_{15}) = 2 \cdot 9.89882 \dots - 19.79795 \dots = - 0.00029 \dots

u_{16} = 9.89890 \dots

Δ u_{16} = ∣ 9.89890 \dots - 9.89882 \dots ∣ = 0.00007 \dots

Δ u_{16} = 0.00007 \dots

u_{17} = 9.89894 \dots : Δ u_{17} = 0.00003 \dots

f (u_{16}) = 9.89890 \dots^{2} - 19.79795 \dots \cdot 9.89890 \dots + 97.98979 \dots = 5.58818 E - 9

f^{^{'}} (u_{16}) = 2 \cdot 9.89890 \dots - 19.79795 \dots = - 0.00014 \dots

u_{17} = 9.89894 \dots

Δ u_{17} = ∣ 9.89894 \dots - 9.89890 \dots ∣ = 0.00003 \dots

Δ u_{17} = 0.00003 \dots

u_{18} = 9.89896 \dots : Δ u_{18} = 0.00001 \dots

f (u_{17}) = 9.89894 \dots^{2} - 19.79795 \dots \cdot 9.89894 \dots + 97.98979 \dots = 1.39724 E - 9

f^{^{'}} (u_{17}) = 2 \cdot 9.89894 \dots - 19.79795 \dots = - 0.00007 \dots

u_{18} = 9.89896 \dots

Δ u_{18} = ∣ 9.89896 \dots - 9.89894 \dots ∣ = 0.00001 \dots

Δ u_{18} = 0.00001 \dots

u_{19} = 9.89897 \dots : Δ u_{19} = 9.35719 E - 6

f (u_{18}) = 9.89896 \dots^{2} - 19.79795 \dots \cdot 9.89896 \dots + 97.98979 \dots = 3.49488 E - 10

f^{^{'}} (u_{18}) = 2 \cdot 9.89896 \dots - 19.79795 \dots = - 0.00003 \dots

u_{19} = 9.89897 \dots

Δ u_{19} = ∣ 9.89897 \dots - 9.89896 \dots ∣ = 9.35719 E - 6

Δ u_{19} = 9.35719 E - 6

u_{20} = 9.89897 \dots : Δ u_{20} = 4.69825 E - 6

f (u_{19}) = 9.89897 \dots^{2} - 19.79795 \dots \cdot 9.89897 \dots + 97.98979 \dots = 8.75531 E - 11

f^{^{'}} (u_{19}) = 2 \cdot 9.89897 \dots - 19.79795 \dots = - 0.00001 \dots

u_{20} = 9.89897 \dots

Δ u_{20} = ∣ 9.89897 \dots - 9.89897 \dots ∣ = 4.69825 E - 6

Δ u_{20} = 4.69825 E - 6

u_{21} = 9.89897 \dots : Δ u_{21} = 2.39033 E - 6

f (u_{20}) = 9.89897 \dots^{2} - 19.79795 \dots \cdot 9.89897 \dots + 97.98979 \dots = 2.20837 E - 11

f^{^{'}} (u_{20}) = 2 \cdot 9.89897 \dots - 19.79795 \dots = - 9.23875 E - 6

u_{21} = 9.89897 \dots

Δ u_{21} = ∣ 9.89897 \dots - 9.89897 \dots ∣ = 2.39033 E - 6

Δ u_{21} = 2.39033 E - 6

u_{22} = 9.89897 \dots : Δ u_{22} = 1.28144 E - 6

f (u_{21}) = 9.89897 \dots^{2} - 19.79795 \dots \cdot 9.89897 \dots + 97.98979 \dots = 5.71276 E - 12

f^{^{'}} (u_{21}) = 2 \cdot 9.89897 \dots - 19.79795 \dots = - 4.45809 E - 6

u_{22} = 9.89897 \dots

Δ u_{22} = ∣ 9.89897 \dots - 9.89897 \dots ∣ = 1.28144 E - 6

Δ u_{22} = 1.28144 E - 6

u_{23} = 9.89897 \dots : Δ u_{23} = 8.62303 E - 7

f (u_{22}) = 9.89897 \dots^{2} - 19.79795 \dots \cdot 9.89897 \dots + 97.98979 \dots = 1.63425 E - 12

f^{^{'}} (u_{22}) = 2 \cdot 9.89897 \dots - 19.79795 \dots = - 1.89521 E - 6

u_{23} = 9.89897 \dots

Δ u_{23} = ∣ 9.89897 \dots - 9.89897 \dots ∣ = 8.62303 E - 7

Δ u_{23} = 8.62303 E - 7

u \approx 9.89897 \dots

Apply long division:

\frac{u ^{2} - 19.79795 \dots u + 97.98979 \dots}{u - 9.89897 \dots} = u - 9.89897 \dots

u - 9.89897 \dots \approx 0

u \approx 9.89897 \dots

Approximate to algebraic form

u = \frac{9701}{980}

The solutions are

u \approx 0.10101 \dots, u \approx 0.10102 \dots, u \approx 9.89897 \dots, u = \frac{9701}{980}

u \approx 0.10101 \dots, u \approx 0.10102 \dots, u \approx 9.89897 \dots, u = \frac{9701}{980}

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

(\frac{u + u ^{- 1}}{2})^{2} - 10 \frac{u + u ^{- 1}}{2} + 25

and compare to zero

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u \approx 0.10101 \dots, u \approx 0.10102 \dots, u \approx 9.89897 \dots, u = \frac{9701}{980}

u \approx 0.10101 \dots, u \approx 0.10102 \dots, u \approx 9.89897 \dots, u = \frac{9701}{980}

Substitute back

u = e^{x},

solve for

x

Solve

e^{x} = 0.10101 \dots : x = ln (0.10101 \dots)

e^{x} = 0.10101 \dots

Apply exponent rules

e^{x} = 0.10101 \dots

f (x) = g (x)

, then

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

ln (e^{x}) = ln (0.10101 \dots)

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (0.10101 \dots)

x = ln (0.10101 \dots)

Solve

e^{x} = 0.10102 \dots : x = ln (0.10102 \dots)

e^{x} = 0.10102 \dots

Apply exponent rules

e^{x} = 0.10102 \dots

f (x) = g (x)

, then

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

ln (e^{x}) = ln (0.10102 \dots)

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (0.10102 \dots)

x = ln (0.10102 \dots)

Solve

e^{x} = 9.89897 \dots : x = ln (9.89897 \dots)

e^{x} = 9.89897 \dots

Apply exponent rules

e^{x} = 9.89897 \dots

f (x) = g (x)

, then

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

ln (e^{x}) = ln (9.89897 \dots)

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (9.89897 \dots)

x = ln (9.89897 \dots)

Solve

e^{x} = \frac{9701}{980} : x = ln (\frac{9701}{980})

e^{x} = \frac{9701}{980}

Apply exponent rules

e^{x} = \frac{9701}{980}

f (x) = g (x)

, then

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

ln (e^{x}) = ln (\frac{9701}{980})

Apply log rule:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (\frac{9701}{980})

x = ln (\frac{9701}{980})

x = ln (0.10101 \dots), x = ln (0.10102 \dots), x = ln (9.89897 \dots), x = ln (\frac{9701}{980})

x = ln (0.10101 \dots), x = ln (0.10102 \dots), x = ln (9.89897 \dots), x = ln (\frac{9701}{980})

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