90-0.04x=30e^{0.001x}

Lösung

90 - 0.04 x = 30 e^{0.001 x}

x = - \frac{0.04 W _{0} ( \frac{0.03 e ^{2.25}}{0.04} ) - 0.09}{0.00004}

Dezimale

x = 715.73409 \dots

Schritte zur Lösung

90 - 0.04 x = 30 e^{0.001 x}

90 - 0.04 x = 30 e^{0.001 x}

für die Lambert-Form vorbereiten:

(90 - 0.04 x) e^{- 0.001 x} = 30

Schreibe die Gleichung um mit

- 0.001 x + \frac{0.09}{0.04} = u

und

x = - \frac{u - 2.25}{0.001}

(90 - 0.04 (- \frac{u - 2.25}{0.001})) e^{- 0.001 (- \frac{u - 2.25}{0.001})} = 30

Vereinfache

e^{u - 2.25} (90 + \frac{0.04 ( u - 2.25 )}{0.001}) = 30

e^{u - 2.25} (90 + \frac{0.04 ( u - 2.25 )}{0.001}) = 30

in Lambert-Form umschreiben:

e^{u} u = \frac{0.03 e ^{2.25}}{0.04}

Löse

e^{u} u = \frac{0.03 e ^{2.25}}{0.04} : u = W_{0} (\frac{0.03 e ^{2.25}}{0.04})

u = W_{0} (\frac{0.03 e ^{2.25}}{0.04})

Setze

u = - 0.001 x + \frac{0.09}{0.04} w i e d er e in,

löse für

x

Löse

- 0.001 x + \frac{0.09}{0.04} = W_{0} (\frac{0.03 e ^{2.25}}{0.04}) : x = - \frac{0.04 W _{0} ( \frac{0.03 e ^{2.25}}{0.04} ) - 0.09}{0.00004}

x = - \frac{0.04 W _{0} ( \frac{0.03 e ^{2.25}}{0.04} ) - 0.09}{0.00004}

5^{n} = 1

12 5^{l o g_{3} (x + 3)} = 8

e^{(2 x)} - 3 e^{x} + 2 = 0

2^{x} - 2^{- x} = 5 (1 - 2^{- x})

3^{x - 2} \cdot 2^{x - 1} = 1125