Integrating Factor for First Order ODE/Function of Product of Variables
Theorem
Let the first order ordinary differential equation:
- $(1): \quad \map M {x, y} + \map N {x, y} \dfrac {\d y} {\d x} = 0$
be non-homogeneous and not exact.
Let $(1)$ be such that:
- $\map g z = \dfrac {\dfrac {\partial M} {\partial y} - \dfrac {\partial N} {\partial x}} {N y - M x}$
is a function of $z$, where $z = x y$.
Then:
- $\map \mu {x y} = \map \mu z = e^{\int \map g z \d z}$
is an integrating factor for $(1)$.
Proof
Preliminary Work
Let us for ease of manipulation express $(1)$ in the form of differentials:
- $(2): \quad \map M {x, y} \rd x + \map N {x, y} \rd y = 0$
Let $\mu$ be an integrating factor for $(2)$.
Then, by definition:
- $\mu \, \map M {x, y} \rd x + \mu \, \map N {x, y} \rd y = 0$
is an exact differential equation.
By Solution to Exact Differential Equation:
- $\dfrac {\map \partial {\mu M} } {\partial y} = \dfrac {\map \partial {\mu N} } {\partial x}$
Evaluating this, using the Product Rule for Derivatives:
- $\mu \dfrac {\partial M} {\partial y} + M \dfrac {\partial \mu} {\partial y} = \mu \dfrac {\partial N} {\partial x} + N \dfrac {\partial \mu} {\partial x}$
which leads us to:
- $\dfrac 1 \mu \paren {N \dfrac {\partial \mu} {\partial x} - M \dfrac {\partial \mu} {\partial y} } = \dfrac {\partial M} {\partial y} - \dfrac {\partial N} {\partial x}$
Let us use $\map P {x, y}$ for $\dfrac {\partial M} {\partial y} - \dfrac {\partial N} {\partial x}$.
Then:
- $(3): \quad \dfrac 1 \mu = \dfrac {\map P {x, y} } {N \dfrac {\partial \mu} {\partial x} - M \dfrac {\partial \mu} {\partial y}}$
Proof for Function of x y
Suppose that $\mu$ is a function of $z = x y$.
Then:
- $\dfrac {\partial z} {\partial x} = y$
- $\dfrac {\partial z} {\partial y} = x$
Thus:
- $\dfrac {\partial \mu} {\partial x} = \dfrac {\d \mu} {\d z} \dfrac {\partial z} {\partial x} = y \dfrac {\d \mu} {\d z}$
- $\dfrac {\partial \mu} {\partial y} = \dfrac {\d \mu} {\d z} \dfrac {\partial z} {\partial y} = x \dfrac {\d \mu} {\d z}$
which, when substituting in $(3)$, leads us to:
- $\dfrac 1 \mu \dfrac {\d \mu} {\d z} = \dfrac {\map P {x, y} } {N y - M x} = \map g z$
where $\map g z$ is the function of $z$ that we posited.
Conclusion of Proof
We have an equation of the form:
- $\dfrac 1 \mu \dfrac {\d \mu} {\d w} = \map f w$
which is what you get when you apply the Chain Rule for Derivatives and Derivative of Logarithm Function to:
- $\dfrac {\map \d {\ln \mu} } {\d w} = \map f w$
Thus:
- $\ds \ln \mu = \int \map f w \rd w$
and so:
- $\mu = e^{\int \map f w \rd w}$
Hence the results as stated.
$\blacksquare$
Sources
- 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 2.9$: Integrating Factors: Problem $1$