# Bernoulli differential equation

:"This topic in

mathematics is named afterJakob Bernoulli . SeeBernoulli's principle for an unrelated topic influid dynamics , named after the inventorDaniel Bernoulli ."In

mathematics , anordinary differential equation of the form:$y\text{'}+\; P(x)y\; =\; Q(x)y^n,$

is called a

**Bernoulli differential equation**or**Bernoulli equation**when n≠1, 0. Dividing by $y^n$ yields:$frac\{y\text{'}\}\{y^\{n\; +\; frac\{P(x)\}\{y^\{n-1\; =\; Q(x).$Achange of variables is made to transform into a linear first-order differential equation.:$w=frac\{1\}\{y^\{n-1$:$w\text{'}=frac\{(1-n)\}\{y^\{ny\text{'}$:$frac\{w\text{'}\}\{1-n\}\; +\; P(x)w\; =\; Q(x)$The substituted equation can be solved using the

integrating factor :$M(x)=\; e^\{(1-n)int\; P(x)dx\}.$

**Example**Consider the Bernoulli equation :$y\text{'}\; -\; frac\{2y\}\{x\}\; =\; -x^2y^2$Division by $y^2$ yields:$y\text{'}y^\{-2\}\; -\; frac\{2\}\{x\}y^\{-1\}\; =\; -x^2$Changing variables gives the equations:$w\; =\; frac\{1\}\{y\}$:$w\text{'}\; =\; frac\{-y\text{'}\}\{y^2\}.$:$w\text{'}\; +\; frac\{2\}\{x\}w\; =\; x^2$which can be solved using the integrating factor:$M(x)=\; e^\{2int\; frac\{1\}\{x\}dx\}\; =\; x^2.$Multiplying by $M(x)$,:$w\text{'}x^2\; +\; 2xw\; =\; x^4,,$ Note that left side is the

derivative of $wx^2$. Integrating both sides results in the equations:$int\; (wx^2)\text{'}\; dx\; =\; int\; x^4\; dx$:$wx^2\; =\; frac\{1\}\{5\}x^5\; +\; C$ :$frac\{1\}\{y\}x^2\; =\; frac\{1\}\{5\}x^5\; +\; C$The solution for $y$ is:$y\; =\; frac\{x^2\}\{frac\{1\}\{5\}x^5\; +\; C\}$**External links***

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