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Bernoulli鈥檚 equation

A first-order equation that can be written in the form $\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{+}\mathbf{P}\left(\mathbf{x}\right)\mathbf{y}\mathbf{=}\mathbf{Q}\left(\mathbf{x}\right){\mathbf{y}}^{\mathbf{n}}$, where P(x) and Q(x) are continuous on an interval (a,b) and n is a real number, is called a Bernoulli equation.

Given, $\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{-}\mathbf{y}\mathbf{=}{\mathbf{e}}^{\mathbf{2}\mathbf{x}}{\mathbf{y}}^{\mathbf{3}}$.

Compare with the general form of the Bernoulli equation.

n = 3, $\mathbf{P}\left(\mathbf{x}\right)\mathbf{=}\mathbf{-}\mathbf{1}$, and $\mathbf{Q}\left(\mathbf{x}\right)\mathbf{=}{\mathbf{e}}^{\mathbf{2}\mathbf{x}}$.

Now, divide by ${\mathbf{y}}^{\mathbf{3}}$, we get,

${\mathbf{y}}^{\mathbf{-}\mathbf{3}}\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{-}{\mathbf{y}}^{\mathbf{-}\mathbf{2}}\mathbf{=}{\mathbf{e}}^{\mathbf{2}\mathbf{x}}路路路路路路\left(1\right)$

Substitute $\mathbf{v}\mathbf{=}{\mathbf{y}}^{\mathbf{-}\mathbf{2}}$.

Differentiate with respect to x.

$\begin{array}{c}\frac{\mathbf{dv}}{\mathbf{dx}}\mathbf{=}\mathbf{-}\mathbf{2}{\mathbf{y}}^{\mathbf{-}\mathbf{3}}\frac{\mathbf{dy}}{\mathbf{dx}}\\ \mathbf{-}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{dv}}{\mathbf{dx}}\mathbf{=}{\mathbf{y}}^{\mathbf{-}\mathbf{3}}\frac{\mathbf{dy}}{\mathbf{dx}}\end{array}$

Substitute it on equation (1)

$\mathbf{-}\frac{\mathbf{1}}{\mathbf{2}}\frac{\mathbf{dv}}{\mathbf{dx}}\mathbf{-}\mathbf{v}\mathbf{=}{\mathbf{e}}^{\mathbf{2}\mathbf{x}}$

$\frac{\mathbf{dv}}{\mathbf{dx}}\mathbf{+}\mathbf{2}\mathbf{v}\mathbf{=}\mathbf{-}\mathbf{2}{\mathbf{e}}^{\mathbf{2}\mathbf{x}}路路路路路路\left(2\right)$

Now, integrate P(x) the first. Where $\mathbf{P}\left(\mathbf{x}\right)\mathbf{=}\mathbf{2}$.

$\begin{array}{c}鈭�\mathbf{P}\left(\mathbf{x}\right)\mathbf{dx}\mathbf{=}鈭�\mathbf{2}\mathbf{dx}\\ \mathbf{=}\mathbf{2}\mathbf{x}\end{array}$

Then,

$\begin{array}{c}渭\left(\mathbf{x}\right)\mathbf{=}{\mathbf{e}}^{鈭�\mathbf{P}\left(\mathbf{x}\right)\mathbf{dx}}\\ \mathbf{=}{\mathbf{e}}^{\mathbf{2}\mathbf{x}}\end{array}$

Multiply $渭\left(\mathbf{x}\right)$with equation (2).

$\begin{array}{c}{\mathbf{e}}^{\mathbf{2}\mathbf{x}}\frac{\mathbf{dv}}{\mathbf{dx}}\mathbf{+}\mathbf{2}{\mathbf{e}}^{\mathbf{2}\mathbf{x}}\mathbf{v}\mathbf{=}\mathbf{-}\mathbf{2}{\mathbf{e}}^{\mathbf{4}\mathbf{x}}\\ \frac{\mathbf{d}}{\mathbf{dx}}\left[{\mathbf{e}}^{\mathbf{2}\mathbf{x}}\mathbf{v}\right]\mathbf{=}\mathbf{-}\mathbf{2}{\mathbf{e}}^{\mathbf{4}\mathbf{x}}\end{array}$

Integrate both sides,

$\begin{array}{l}鈭�\frac{\mathbf{d}}{\mathbf{dx}}\left[{\mathbf{e}}^{\mathbf{2}\mathbf{x}}\mathbf{v}\right]\mathbf{dx}\mathbf{=}\mathbf{-}\mathbf{2}鈭�{\mathbf{e}}^{\mathbf{4}\mathbf{x}}\mathbf{dx}\\ {\mathbf{e}}^{\mathbf{2}\mathbf{x}}\mathbf{v}\mathbf{=}\mathbf{-}\mathbf{2}鈭�{\mathbf{e}}^{\mathbf{4}\mathbf{x}}\mathbf{dx}\\ {\mathbf{e}}^{\mathbf{2}\mathbf{x}}\mathbf{v}\mathbf{=}\mathbf{-}\frac{{\mathbf{e}}^{\mathbf{4}\mathbf{x}}}{\mathbf{2}}\mathbf{+}{\mathbf{C}}_{\mathbf{1}}\\ \mathbf{v}\mathbf{=}\mathbf{-}\frac{{\mathbf{e}}^{\mathbf{2}\mathbf{x}}}{\mathbf{2}}\mathbf{+}{\mathbf{e}}^{\mathbf{-}\mathbf{2}\mathbf{x}}{\mathbf{C}}_{\mathbf{1}}\\ \mathbf{v}\mathbf{=}\frac{\mathbf{2}{\mathbf{e}}^{\mathbf{-}\mathbf{2}\mathbf{x}}{\mathbf{C}}_{\mathbf{1}}\mathbf{-}{\mathbf{e}}^{\mathbf{2}\mathbf{x}}}{\mathbf{2}}\\ \mathbf{v}\mathbf{=}\frac{{\mathbf{e}}^{\mathbf{-}\mathbf{2}\mathbf{x}}\mathbf{C}\mathbf{-}{\mathbf{e}}^{\mathbf{2}\mathbf{x}}}{\mathbf{2}}\end{array}$

Substitute $\mathbf{v}\mathbf{=}{\mathbf{y}}^{\mathbf{-}\mathbf{2}}$.

$\begin{array}{c}{\mathbf{y}}^{\mathbf{-}\mathbf{2}}\mathbf{=}\frac{{\mathbf{e}}^{\mathbf{-}\mathbf{2}\mathbf{x}}\mathbf{C}\mathbf{-}{\mathbf{e}}^{\mathbf{2}\mathbf{x}}}{\mathbf{2}}\\ {\mathbf{y}}^{\mathbf{2}}\mathbf{=}\frac{\mathbf{2}}{{\mathbf{e}}^{\mathbf{-}\mathbf{2}\mathbf{x}}\mathbf{C}\mathbf{-}{\mathbf{e}}^{\mathbf{2}\mathbf{x}}}\\ \mathbf{y}\mathbf{=}\mathbf{卤}\sqrt{\frac{\mathbf{2}}{{\mathbf{e}}^{\mathbf{-}\mathbf{2}\mathbf{x}}\mathbf{C}\mathbf{-}{\mathbf{e}}^{\mathbf{2}\mathbf{x}}}}\end{array}$

Hence, the solution is$\mathbf{y}\mathbf{=}\mathbf{卤}\sqrt{\frac{\mathbf{2}}{{\mathbf{e}}^{\mathbf{-}\mathbf{2}\mathbf{x}}\mathbf{C}\mathbf{-}{\mathbf{e}}^{\mathbf{2}\mathbf{x}}}}$