Integrating Factors Found by Inspection

The following are the four exact differentials that occurs frequently.

  1.   $d(xy) = x \, dy + y \, dx$
     
  2.   $d\left( \dfrac{x}{y} \right) = \dfrac{y \, dx - x \, dy}{y^2}$
     
  3.   $d\left( \dfrac{y}{x} \right) = \dfrac{x \, dy - y \, dx}{x^2}$
     
  4.   $d\left( \arctan \dfrac{y}{x} \right) = \dfrac{x \, dy - y \, dx}{x^2 + y^2}$
  5.  

  6.   $d\left( \arctan \dfrac{x}{y} \right) = \dfrac{y \, dx - x \, dy}{x^2 + y^2}$

 

The Determination of Integrating Factor

From the differential equation
 

$M ~ dx + N ~ dy = 0$

 

Rule 1
If   $\dfrac{1}{N}\left( \dfrac{\partial M}{\partial y} - \dfrac{\partial N}{\partial x} \right) = f(x)$,   a function of x alone, then   $u = e^{\int f(x)~dx}$   is the integrating factor.

 

Rule 2
If   $\dfrac{1}{M}\left( \dfrac{\partial M}{\partial y} - \dfrac{\partial N}{\partial x} \right) = f(y)$,   a function of y alone, then   $u = e^{-\int f(y)~dy}$   is the integrating factor.

 

Note that the above criteria is of no use if the equation does not have an integrating factor that is a function of x alone or y alone.
 

Steps

  1. Take the coefficient of dx as M and the coefficient of dy as N.
  2. Evaluate M/y and N/x.
  3. Take the difference M/y - N/x.
  4. Divide the result of Step 3 by N. If the quotient is a function of x alone, use the integrating factor defined in Rule 1 above and proceed to Step 6. If the quotient is not a function of x alone, proceed to Step 5.
  5. Divide the result of Step 3 by M. If the quotient is a function of y alone, use the integrating factor defined in Rule 2 above and proceed to Step 6. If the quotient is not a function of y alone, look for another method of solving the equation.
  6. Multiply both sides of the given equation by the integrating factor u, the new equation which is uM dx + uN dy = 0 should be exact.
  7. Solve the result of Step 6 by exact equation or by inspection.

 

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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