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.

 

Example 2

$(3 + x\cos y) ~ dx - x^2 \sin y ~ dy$
 

The quantity (-sin y dy) is the exact derivative of cos y. Let
$z = \cos y$

$dz = -\sin y ~ dy$
 

Substitute,
$(3 + xz) ~ dx + x^2 ~ dz$

then continue solving.

 

Bernoulli's Equation
Bernoulli's equation is in the form
 

$dy + P(x)~y~dx = Q(x)~y^n~dx$

 

If x is the dependent variable, Bernoulli's equation can be recognized in the form   $dx + P(y)~x~dy = Q(y)~x^n~dy$.
 

If n = 1, the variables are separable.
If n = 0, the equation is linear.
If n ≠ 1, Bernoulli's equation.
 

Steps in solving Bernoulli's equation

  1. Write the equation into the form   $dy + Py~dx = Qy^n~dx$.
     
  2. Identify   $P$,   $Q$,   and   $n$.
     
  3. Write the quantity   $(1 - n)$   and let   $z = y^{(1 - n)}$.
     
  4. Determine the integrating factor   $u = e^{(1 - n)\int P~dx}$.
     
  5. The solution is defined by   $\displaystyle zu = (1 - n)\int Qu~dx + C$.
     
  6. Bring the result back to the original variable.

 

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