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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 and is a real number, is called a Bernoulli equation.

Given, $\frac{\mathbf{dr}}{\mathbf{d}胃}\mathbf{=}\frac{{\mathbf{r}}^{\mathbf{2}}\mathbf{+}\mathbf{2}\mathbf{r}胃}{{胃}^{\mathbf{2}}}$.

Evaluate it.

$\begin{array}{c}\frac{\mathbf{dr}}{\mathbf{d}胃}\mathbf{=}\frac{{\mathbf{r}}^{\mathbf{2}}\mathbf{+}\mathbf{2}\mathbf{r}胃}{{胃}^{\mathbf{2}}}\\ \mathbf{=}\frac{{\mathbf{r}}^{\mathbf{2}}}{{胃}^{\mathbf{2}}}\mathbf{+}\frac{\mathbf{2}\mathbf{r}}{胃}\\ \frac{\mathbf{dr}}{\mathbf{d}胃}\mathbf{-}\frac{\mathbf{2}}{胃}\mathbf{r}\mathbf{=}{\mathbf{r}}^{\mathbf{2}}{胃}^{\mathbf{-}\mathbf{2}}\end{array}$

Compare with the general form of the Bernoulli equation.

n = 2, $\mathbf{P}\left(胃\right)\mathbf{=}\mathbf{-}\frac{\mathbf{2}}{胃}$, and $\mathbf{Q}\left(胃\right)\mathbf{=}{胃}^2$.

Now, divide by , we get,

${\mathbf{r}}^{\mathbf{-}\mathbf{2}}\frac{\mathbf{dr}}{\mathbf{d}胃}\mathbf{-}\frac{\mathbf{2}}{胃}{\mathbf{r}}^{\mathbf{-}\mathbf{1}}\mathbf{=}{胃}^{\mathbf{-}\mathbf{2}}路路路路路路\left(1\right)$

Substitute $\mathbf{v}\mathbf{=}{\mathbf{r}}^{\mathbf{-}\mathbf{1}}$.

Differentiate with respect to $胃$.

$\begin{array}{c}\frac{\mathbf{dv}}{\mathbf{d}胃}\mathbf{=}\mathbf{-}{\mathbf{r}}^{\mathbf{-}\mathbf{2}}\frac{\mathbf{dr}}{\mathbf{d}胃}\\ \mathbf{-}\frac{\mathbf{dv}}{\mathbf{d}胃}\mathbf{=}{\mathbf{r}}^{\mathbf{-}\mathbf{2}}\frac{\mathbf{dr}}{\mathbf{d}胃}\end{array}$

Substitute it on equation (1)

$\begin{array}{l}-\frac{\mathbf{dv}}{\mathbf{d}胃}-\frac{2}{胃}\mathbf{v}\mathbf{=}{胃}^{-2}\\ \frac{\mathbf{dv}}{\mathbf{d}胃}\mathbf{+}\frac{2}{胃}\mathbf{v}\mathbf{=}-{胃}^{-2}路路路路路路\left(2\right)\end{array}$

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

$\begin{array}{c}鈭�\mathbf{P}\left(胃\right)\mathbf{d}胃\mathbf{=}\mathbf{2}鈭�\frac{\mathbf{1}}{胃}\mathbf{d}胃\\ \mathbf{=}\mathbf{2}\mathbf{ln}\left|胃\right|\end{array}$

Then,

$\begin{array}{c}渭\left(胃\right)\mathbf{=}{\mathbf{e}}^{鈭�\mathbf{P}\left(胃\right)\mathbf{d}胃}\\ \mathbf{=}{\mathbf{e}}^{\mathbf{2}\mathbf{ln}\left|胃\right|}\\ \mathbf{=}{胃}^{\mathbf{2}}\end{array}$

Multiply $渭\left(胃\right)$with equation (2).

$\begin{array}{c}{胃}^{\mathbf{2}}\frac{\mathbf{dv}}{\mathbf{d}胃}\mathbf{+}\frac{\mathbf{2}}{胃}{胃}^{\mathbf{2}}\mathbf{v}\mathbf{=}\mathbf{-}\mathbf{1}\\ {胃}^{\mathbf{2}}\frac{\mathbf{dv}}{\mathbf{d}胃}\mathbf{+}\mathbf{2}胃\mathbf{v}\mathbf{=}\mathbf{-}\mathbf{1}\\ \frac{\mathbf{d}}{\mathbf{d}胃}\left[{胃}^{\mathbf{2}}\mathbf{v}\right]\mathbf{=}\mathbf{-}\mathbf{1}\end{array}$

Integrate both sides,

$\begin{array}{c}鈭�\frac{\mathbf{d}}{\mathbf{d}胃}\left[{胃}^{\mathbf{2}}\mathbf{v}\right]\mathbf{d}胃\mathbf{=}鈭�\mathbf{-}\mathbf{1}\mathbf{d}胃\\ {胃}^{\mathbf{2}}\mathbf{v}\mathbf{=}\mathbf{-}胃\mathbf{+}{\mathbf{C}}_{\mathbf{1}}\\ \mathbf{v}\mathbf{=}\frac{\mathbf{C}\mathbf{-}胃}{{胃}^{\mathbf{2}}}\end{array}$

Substitute $\mathbf{v}\mathbf{=}{\mathbf{r}}^{\mathbf{-}\mathbf{1}}$.

$\begin{array}{c}{\mathbf{r}}^{\mathbf{-}\mathbf{1}}\mathbf{=}\frac{\mathbf{C}\mathbf{-}胃}{{胃}^{\mathbf{2}}}\\ \mathbf{r}\mathbf{=}\frac{{胃}^{\mathbf{2}}}{\mathbf{C}\mathbf{-}胃}\end{array}$

Hence the solution is$\mathbf{r}\mathbf{=}\frac{{胃}^{\mathbf{2}}}{\mathbf{C}\mathbf{-}胃}$