Integrating Factor for First Order ODE/Conclusion
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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 $\map \mu {x, y}$be an integrating factor for $(1)$.
If one of these is the case:
- $\mu$ is a function of $x$ only
- $\mu$ is a function of $y$ only
- $\mu$ is a function of $x + y$
- $\mu$ is a function of $x y$
then:
- $\mu = e^{\int \map f w \rd w}$
where $w$ depends on the nature of $\mu$.
Proof
We have one of these:
- Integrating Factor for First Order ODE: Function of One Variable: $x$ or $y$ only
- Integrating Factor for First Order ODE: Function of $x + y$
- Integrating Factor for First Order ODE: Function of $x y$
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