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+xcosy) dx−x2siny dy
 

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

dz=−siny dy
 

Substitute,
(3+xz) dx+x2 dz

then continue solving.

 

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

dy+P(x) y dx=Q(x) yn dx

 

If x is the dependent variable, Bernoulli's equation can be recognized in the form   dx+P(y) x dy=Q(y) xn 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=Qyn 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)∫P dx.
     
  5. The solution is defined by   zu=(1−n)∫Qu dx+C.
     
  6. Bring the result back to the original variable.

 

Tags
substitution
linear differential equation
integrating factor
Bernoulli's equation
suggested substitution