## Integrating Factors Found by Inspection

The following are the four exact differentials that occurs frequently.

- $d(xy) = x \, dy + y \, dx$

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

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

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

## The Determination of Integrating Factor

From the differential equation

**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**

- Take the coefficient of
*dx*as*M*and the coefficient of*dy*as*N*. - Evaluate ∂
*M*/∂*y*and ∂*N*/∂*x*. - Take the difference ∂
*M*/∂*y*- ∂*N*/∂*x*. - 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. - 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. - 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. - Solve the result of Step 6 by exact equation or by inspection.

## Substitution Suggested by the Equation

**Example 1**

The quantity (2*x* - *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**

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

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**

- Write the equation into the form $dy + Py~dx = Qy^n~dx$.

- Identify $P$, $Q$, and $n$.

- Write the quantity $(1 - n)$ and let $z = y^{(1 - n)}$.

- Determine the integrating factor $u = e^{(1 - n)\int P~dx}$.

- The solution is defined by $\displaystyle zu = (1 - n)\int Qu~dx + C$.

- Bring the result back to the original variable.