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1. The problem statement, all variables and given/known data

Solve the ODE y' + (3/t) y = t^{3}.

2. Relevant equations/concepts

1st order linear ODE

3. The attempt at a solution

Integrating factor

=exp ∫(3/t)dt

=exp (3ln|t| + k)

=exp (ln|t|^{3}) (take constant of integration k=0)

=|t|^{3}

If the integrating factor were |t|^{2}= t^{2}, I wouldn't have any problem with it, the absolute value is gone, luckily.

But now here in this case, multiplying both sides of the ODE by |t|^{3}, the absolute value is giving me trouble. How can I proceed? Can I just forget about the absolute value and mutliply the ODE simply by t^{3}? (I've seen a lot of people doing this, but I don't think it's correct...)

I am never able to understand how to deal with problems like this. What is the correct way to handle these problems where there is an absolute value sign in the integrating factor?

Any help in this matter is greatly appreciated!

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# First order linear ODE-integrating factor has absolute value in it

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