Bernoulli's equation

 
 

Problem 04 | Bernoulli's Equation

Problem 04
$y' = y - xy^3e^{-2x}$
 

Solution 04
$y' = y - xy^3e^{-2x}$

$\dfrac{dy}{dx} - y = -xe^{-2x}y^3$

$dy - y~dx = -xe^{-2x}y^3~dx$       → Bernoulli's equation

$dy + Py~dx = Qy^n~dx$
 

From which
$P = -1$

$Q = -xe^{-2x}$

$n = 3$

$(1 - n) = -2$

$z = y^{1 - n} = y^{-2}$
 

Integrating factor,
$u = e^{(1 - n)\int P~dx} = e^{-2\int (-1)~dx}$

$u = e^{2\int dx} = e^{2x}$
 

Thus,
$\displaystyle zu = (1 - n)\int Qu~dx + C$

Substitution Suggested by the Equation | Bernoulli's Equation

Substitution Suggested by the Equation
Example 1

$(2x - y + 1)~dx - 3(2x - y)~dy = 0$
 

The quantity (2x - y) appears twice in the equation. Let
$z = 2x - y$

$dz = 2~dx - dy$

$dy = 2~dx - dz$
 

Substitute,
$(z + 1)~dx - 3z(2~dx - dz) = 0$

then continue solving.

 

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