Question
Explicitly show that the general Bernoulli equation (4.2) can be transformed into a linear ODE by making use of the transformation $p=1 / y^{n-1}$. Here $n$ is an arbitrary positive integer greater than or equal to two.
Step 1
Step 1: Start by writing down the general form of the Bernoulli equation, which is given by: \[ y' + p(x) y = q(x) y^n \] where \( n \) is a positive integer, \( n \geq 2 \), and \( p(x) \) and \( q(x) \) are functions of \( x \). Show more…
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A Bernoulli differential equation (named after James Bernoulli) is of the form $ \frac {dy}{dx} + P(x)y = Q(x)y^n $ Observe that, if $ n = 0 $ or 1, the Bernoulli equation is linear. For other values of $ n, $ show that the substitution $ u = y^{1-n} $ transforms the Bernoulli equation into the linear equation $ \frac {du}{dx} + ( 1 - n)P(x)u = (1 - n)Q(x) $
Differential Equations
Linear Equations
Observe that, if $n=0$ or 1 , the Bernoulli equation is linear. For other values of $n$, show that the substitution $u=y^{1}$ * transforms the Bernoulli equation into the linear equation $$ \frac{d u}{d x}+(1-n) P(x) u=(1-n) Q(x) $$
A Bernoulli differential equation (named after James Bernoulli) is of the form $$\frac{d y}{d x}+P(x) y=Q(x) y^{n}$$ Observe that, if $n=0$ or $1,$ the Bernoulli equation is linear. For other values of $n$ , show that the substitution $u=y^{1-n}$ transforms the Bernoulli equation into the linear equation $$\frac{d u}{d x}+(1-n) P(x) u=(1-n) Q(x)$$
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