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受欢迎的代数 >

e^2x-5e^x+6=0

解答

e^{2} x - 5 e^{x} + 6 = 0

解答

x = - \frac{e ^{2} W _{- 1} ( - \frac{5}{e ^{\frac{2 e ^{2} + 6}{e ^{2}}}} ) + 6}{e ^{2}}, x = - \frac{e ^{2} W _{0} ( - \frac{5}{e ^{\frac{2 e ^{2} + 6}{e ^{2}}}} ) + 6}{e ^{2}}

+1

十进制

x = 0.96613 \dots, x = - 0.32126 \dots

求解步骤

e^{2} x - 5 e^{x} + 6 = 0

将

e^{2} x - 5 e^{x} + 6 = 0

调整为朗伯形式:

- 5 = - e^{- x} (e^{2} x + 6)

用

- x - \frac{6}{e ^{2}} = u

和

x = - \frac{e ^{2} u + 6}{e ^{2}}

改写方程式

- 5 = - e^{- (- \frac{e ^{2} u + 6}{e ^{2}})} (e^{2} (- \frac{e ^{2} u + 6}{e ^{2}}) + 6)

化简

- 5 = e^{2} e^{\frac{e ^{2} u + 6}{e ^{2}}} u

改写

- 5 = e^{2} e^{\frac{e ^{2} u + 6}{e ^{2}}} u

为朗伯形式:

e^{u} u = - \frac{5}{e ^{\frac{2 e ^{2} + 6}{e ^{2}}}}

解

e^{u} u = - \frac{5}{e ^{\frac{2 e ^{2} + 6}{e ^{2}}}} : u = W_{- 1} (- \frac{5}{e ^{\frac{2 e ^{2} + 6}{e ^{2}}}}), u = W_{0} (- \frac{5}{e ^{\frac{2 e ^{2} + 6}{e ^{2}}}})

u = W_{- 1} (- \frac{5}{e ^{\frac{2 e ^{2} + 6}{e ^{2}}}}), u = W_{0} (- \frac{5}{e ^{\frac{2 e ^{2} + 6}{e ^{2}}}})

代回

u = - x - \frac{6}{e ^{2}}

，求解

x

解

- x - \frac{6}{e ^{2}} = W_{- 1} (- \frac{5}{e ^{\frac{2 e ^{2} + 6}{e ^{2}}}}) : x = - \frac{e ^{2} W _{- 1} ( - \frac{5}{e ^{\frac{2 e ^{2} + 6}{e ^{2}}}} ) + 6}{e ^{2}}

解

- x - \frac{6}{e ^{2}} = W_{0} (- \frac{5}{e ^{\frac{2 e ^{2} + 6}{e ^{2}}}}) : x = - \frac{e ^{2} W _{0} ( - \frac{5}{e ^{\frac{2 e ^{2} + 6}{e ^{2}}}} ) + 6}{e ^{2}}

x = - \frac{e ^{2} W _{- 1} ( - \frac{5}{e ^{\frac{2 e ^{2} + 6}{e ^{2}}}} ) + 6}{e ^{2}}, x = - \frac{e ^{2} W _{0} ( - \frac{5}{e ^{\frac{2 e ^{2} + 6}{e ^{2}}}} ) + 6}{e ^{2}}

作图

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