Consider the ODE y' + tan(ax)y = sin(ax), where a is a nonzero real number. Find an integrating factor without using an exponential, but instead, a trig function. Then solve. If you used the standard method for finding an integratin factor, would anything in your solution have changed?

Consider the ODE y' + tan(ax)y = sin(ax), where a is a nonzero real number. Find an integrating factor without using an exponential, but instead, a trig function. Then solve. If you used the standard method for finding an integratin factor, would anything in your solution have changed?

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

I'm having a hard time finding an integrating factor without using an exponential. How do I find an integrating factor using a trig function? Please, I've asked this question many times and people keep solving it using the regular I.F. = e^∫p(x)dx and this is NOT what I need.

(a) Consider the ODE y' + tan(ax)y = sin(ax), where a is a nonzero real number. Find an
integrating factor without using an exponential, but instead, a trig function. Then solve.
If
you
used the standard method for finding an integratin factor, would anything in your
solution have changed?
(b) Repeat the steps in part (a) for the ODE y' + cot(ax)y – sin(ax) = 0.
(c) Do you see a pattern? Can you come up with another example of a linear ODE that can
be solved with an integrating factor found without using the standard formula? If so,
does using the standard formula result in the same integrating factor?

With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.

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