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When solving linear ODE of form: $$ y' + p(t)y = g(t) $$

we are looking for an integrating factor u(t), which is given by a formula:

$$ u(t) = \int p(t)dt $$

Now a solution to this form of ODE is:

$$ y(t) = \frac {1}{u(t)} \int_{t_1}^{t_2} p(v)dv +\frac {c}{u(t)} $$

Now in the textbook I use it says that we switch the variable inside the integral from t to v to show that t is an independent variable, so we choose some convenient integration limits and swap them. I understand how to change variable inside integral, but my question is why? Why do we need to change t to v and how does it help us show that t is independent?

Thank you

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1 Answer 1

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I believe it is done in order to distinguish "dummy" integrating variable from a "real" independent variable of which solution depends.

Also, note that they say "switch variables", i.e. simply reliable them. This has nothing to do with "substitutional change of variables", which is replacing a free variable appearing in an equation by an expression depending on other variables.

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