Hello all! I through a section in my text (by https://www.amazon.com/Advanced-Engineering-Mathematics-Michael-Greenberg/dp/0133214311") on 1st Order linear ODEs. I am understanding the derivation of the integrating factor method pretty well; however, there are some aspects of the mathematics that I am getting hung up on. I have an "engineering" background in calculus and hence I get slowed down by the details sometimes.(adsbygoogle = window.adsbygoogle || []).push({});

In order to find the solution to an ODE of the form:

[tex]y' + p(x)y = q(x)\qquad(1)[/tex]

we first consider thehomogeneous case; i.e.,

[tex]y' + p(x)y = 0\qquad(2)[/tex]

After some hootenanny, we arrive at the solution to (2) given by

[tex]y(x) = Ae^{-\int p(x)\,dx}\qquad(3)[/tex]

Now, in order to solve for the constant 'A' given the initial condition y(a) = b, it says that

"...it is convenient to re-express (3) as

[tex]y(x) = Ae^{-\int_a^x p(\xi)\,d\xi}\qquad(4)[/tex]

which is equivalent to (3) since [itex]\int p(x)\,dx \text{ and } \int_a^x p(\xi)\,d\xi[/itex] differ at most by an additive constant, say D, and the resulting [itex]e^D[/itex] can be absorbed into the arbitrary constant A.Question 1:

I am not sure why he even brought this up? Why would changing the dummy variables cause the two integrals to differ *at all* ?

Moving on to thehomogeneous case:

Basically we wish to multiply (2) by some function of x, say [itex]\mu(x)[/itex] giving

[tex]\mu(x) y' + \mu(x)p(x)y = \mu(x)q(x)\qquad(5)[/tex]

so that the left-hand-side of (5) is a derivative. After much more hootenanny, we arrive at the solution

[tex]y(x) = e^{-\int p(x)\,dx}\left(\int e^{\int p(x)\,dx}q(x)\,dx + C\right)\qquad(6)[/tex]

Now again, if we wish to solve for the constant 'C' given the initial condition y(a) = b, we are again advised to use a different expression of (6), namely

[tex]y(x) = e^{-\int_a^x p(\xi)\,d\xi}\left(\int_a^x e^{\int_a^\xi p(\zeta)\,d\zeta}q(\xi)\,d\xi + C\right)\qquad(7)[/tex]

This is just too much changing of dummy variables for my feeble mind to handle .

Question(s) 2:

There are 3 dummy variables being used here:

[tex]e^{-\int_a^x p(\xi)\,d\xi}\qquad(8)[/tex]

[tex] e^{\int_a^\xi p(\zeta)\,d\zeta}\qquad(9)[/tex]

[tex]q(\xi)\,d\xi\qquad(10).[/tex]

I am not really sure what is happening here. They are using [itex]\xi[/itex] in (8) and (10) as dummy variables and in (9) as an endpoint of the integral.

I don't expect anyone to answer all of this at once, but i would love to start a discussion about it so that I can solidify my understanding of the derivation. It's not enough anymore for me to just be able *to use* the integrating factor, but instead to *understand it.*

Thanks,

Casey

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Integrating Factor (Derivation)

**Physics Forums | Science Articles, Homework Help, Discussion**