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受欢迎的代数 >

e^{2x}=3+x

解答

e^{2 x} = 3 + x

解答

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

+1

十进制

x = 0.64694 \dots, x = - 2.99750 \dots

求解步骤

e^{2 x} = 3 + x

将

e^{2 x} = 3 + x

调整为朗伯形式:

1 = (3 + x) e^{- 2 x}

用

- 2 x - 6 = u

和

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

改写方程式

1 = (3 + (- \frac{u + 6}{2})) e^{- 2 (- \frac{u + 6}{2})}

化简

1 = e^{u + 6} (3 - \frac{u + 6}{2})

改写

1 = e^{u + 6} (3 - \frac{u + 6}{2})

为朗伯形式:

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

解

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

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

代回

u = - 2 x - 6

，求解

x

解

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

解

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

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

作图

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