Bernoulli Equations (1st Order)

In mathematics, an ordinary differential equation is called a Bernoulli differential equation if it is of the form

$y'+P(x)y=Q(x)y^{n},$

where $n$ is a real number. Some authors allow any real $n$, whereas others require that $n$ not be 0 or 1. The equation was first discussed in a work of 1695 by Jacob Bernoulli, after whom it is named. The earliest solution, however, was offered by Gottfried Leibniz, who published his result in the same year and whose method is the one still used today.

Bernoulli equations are special because they are nonlinear differential equations with known exact solutions. A notable special case of the Bernoulli equation is the logistic differential equation.

Transformation to a linear 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 $n\neq 0$ and $n\neq 1$, the substitution $u=y^{1-n}$ reduces any Bernoulli equation to a linear differential equation

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

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

Solution

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0 \in (a, b)} and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\{\begin{array}{ll} z: (a,b) \rightarrow (0, \infty)\ ,&\textrm{if}\ \alpha\in \mathbb{R}\setminus\{1,2\},\\ z: (a,b) \rightarrow \mathbb{R}\setminus\{0\}\ ,&\textrm{if}\ \alpha = 2,\\\end{array}\right.}

be a solution of the linear differential equation

$z'(x)=(1-\alpha )P(x)z(x)+(1-\alpha )Q(x).$

Then we have that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(x) := [z(x)]^{\frac{1}{1-\alpha}}} is a solution of

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y'(x)= P(x)y(x) + Q(x)y^\alpha(x)\ ,\ y(x_0) = y_0 := [z(x_0)]^{\frac{1}{1-\alpha}}.}

And for every such differential equation, for all $\alpha >0$ we have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y\equiv 0} as solution for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_0=0} .

Example

Consider the Bernoulli equation

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y' - \frac{2y}{x} = -x^2y^2}

(in this case, more specifically Riccati's equation). The constant function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=0} is a solution. Division by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y^2} yields

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y'y^{-2} - \frac{2}{x}y^{-1} = -x^2}

Changing variables gives the equations

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} u = \frac{1}{y} \; & , ~ u' = \frac{-y'}{y^2} \\ -u' - \frac{2}{x}u &= - x^2 \\ u' + \frac{2}{x}u &= x^2 \end{align}}

which can be solved using the integrating factor

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M(x)= e^{2\int \frac{1}{x}\,dx} = e^{2\ln x} = x^2.}

Multiplying by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M(x)} ,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u'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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} results in the equations

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \int \left(ux^2\right)' dx &= \int x^4\,dx \\ ux^2 &= \frac{1}{5}x^5 + C \\ \frac{1}{y}x^2 &= \frac{1}{5}x^5 + C \end{align}}

The solution for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = \frac{x^2}{\frac{1}{5}x^5 + C}.}

Resources

• Bernoulli Equations Notes. Produced by Paul Dawkins, Lamar University