Bernoulli differential equation

When $n\; =\; 0$, the differential equation is linear. When $n\; =\; 1$, it is separable. In these cases, standard techniques for solving equations of those forms can be applied. For $n\; \backslash neq\; 0$ and $n\; \backslash neq\; 1$, the substitution $u\; =\; y^\{1-n\}$ reduces any Bernoulli equation to a linear differential equation

: $\backslash frac\{du\}\{dx\}\; -\; (n-1)P(x)u\; =\; -\; (n-1)Q(x).$

For example, in the case $n\; =\; 2$, making the substitution $u=y^\{-1\}$ in the differential equation $\backslash frac\{dy\}\{dx\}\; +\; \backslash frac\{1\}\{x\}y=xy^2$ produces the equation $\backslash frac\{du\}\{dx\}\; -\backslash frac\{1\}\{x\}u=-x$, which is a linear differential equation.

Let $x\_0\; \backslash in\; (a,\; b)$ and

:$\backslash begin\{cases\}$

z: (a,b) \rightarrow (0, \infty), &\text{if } \alpha\in \mathbb{R} \smallsetminus \{1,2\},\\[4pt]

z: (a,b) \rightarrow \mathbb{R}\smallsetminus\{0\}, & \text{if } \alpha = 2, \end{cases}

be a solution of the linear differential equation

:$z\text{'}(x)=(1-\backslash alpha)P(x)z(x)\; +\; (1-\backslash alpha)Q(x).$

Then we have that $y(x)\; :=\; [z(x)]^\{1/(1-\backslash alpha)\}$ is a solution of

:$y\text{'}(x)=\; P(x)y(x)\; +\; Q(x)y^\backslash alpha(x)\backslash \; ,\backslash \; y(x\_0)\; =\; y\_0\; :=\; [z(x\_0)]^\{1/(1-\backslash alpha)\}.$

And for every such differential equation, for all $\backslash alpha>0$ we have $y\backslash equiv\; 0$ as solution for $y\_0=0$.

Consider the Bernoulli equation

:$y\text{'}\; -\; \backslash frac\{2y\}\{x\}\; =\; -x^2y^2$

(in this case, more specifically a Riccati equation).

The constant function $y=0$ is a solution.

Division by $y^2$ yields

:$y\text{'}y^\{-2\}\; -\; \backslash frac2x\; y^\{-1\}\; =\; -x^2$

Changing variables gives the equations

:$\backslash begin\{align\}$

u = \frac{1}{y} \; & , ~ u' = \frac{-y'}{y^2} \\[5pt]

-u' - \frac{2}{x}u &= - x^2 \\[5pt]

u' + \frac{2}{x}u &= x^2

\end{align}

which can be solved using the integrating factor

:$M(x)=\; e^\{2\backslash int\; \backslash frac\{1\}\{x\}\backslash ,dx\}\; =\; e^\{2\backslash ln\; x\}\; =\; x^2.$

Multiplying by {{nowrap|$M(x)$,}}

:$u\text{'}x^2\; +\; 2xu\; =\; x^4.$

The left side can be represented as the derivative of $ux^2$ by reversing the product rule. Applying the chain rule and integrating both sides with respect to $x$ results in the equations

:$\backslash begin\{align\}$

\int \left(ux^2\right)' dx &= \int x^4\,dx \\[5pt]

ux^2 &= \frac{1}{5}x^5 + C \\[5pt]

\frac{1}{y}x^2 &= \frac{1}{5}x^5 + C

\end{align}

The solution for $y$ is

:$y\; =\; \backslash frac\{x^2\}\{\backslash frac\{1\}\{5\}x^5\; +\; C\}.$

{{reflist|refs=

{{eom |title=Bernoulli equation |author-first=N. Kh. |author-last=Rozov |oldid=40764}}

{{cite book |last=Zill |first=Dennis G. |date=2013 |title=A First Course in Differential Equations with Modeling Applications |edition=10th |url=https://ebooks.cenreader.com/#!/reader/3007ca4c-3ef6-4b48-a8b3-70ff21126fcf/page/3387eee2-5ffb-4d98-84b1-6402424855a5 |url-access=subscription |location=Boston, Massachusetts |publisher=Cengage Learning |page=73 |isbn=9780357088364}}

{{cite book |last=Stewart |first=James |author-link=James Stewart (mathematician) |date=2015 |title=Calculus: Early Transcendentals |edition=8th |url=https://play.google.com/books/reader?id=hEF-BAAAQBAJ&pg=GBS.PA625 |url-access=subscription |location=Boston, Massachusetts |publisher=Cengage Learning |page=625 |isbn=9781305482463}}

{{cite book |last=Teschl |first=Gerald |author-link=Gerald Teschl |date=2012 |title=Ordinary Differential Equations and Dynamical Systems |url=https://www.mat.univie.ac.at/~gerald/ftp/book-ode/ |location=Providence, Rhode Island |publisher=American Mathematical Society |series=Graduate Studies in Mathematics |issn=1065-7339 |eissn=2376-9203 |page=15 |section=1.4. Finding explicit solutions |section-url=https://www.mat.univie.ac.at/~gerald/ftp/book-ode/ode.pdf#page=24 |isbn=978-0-8218-8328-0 |zbl=1263.34002 |language=en}}

}}

• {{Citation | last1=Bernoulli | first1=Jacob | author1-link=Jacob Bernoulli | title=Explicationes, Annotationes & Additiones ad ea, quae in Actis sup. de Curva Elastica, Isochrona Paracentrica, & Velaria, hinc inde memorata, & paratim controversa legundur; ubi de Linea mediarum directionum, alliisque novis | year=1695 | journal=Acta Eruditorum}}. Cited in {{harvtxt|Hairer|Nørsett|Wanner|1993}}.
• {{Citation | last1=Hairer | first1=Ernst | last2=Nørsett | first2=Syvert Paul | last3=Wanner | first3=Gerhard | title=Solving ordinary differential equations I: Nonstiff problems | publisher=Springer-Verlag | location=Berlin, New York | isbn=978-3-540-56670-0 | year=1993}}.

• [https://planetmath.org/indexofdifferentialequations Index of differential equations]

Category:Ordinary differential equations