Here's an exercise from Tenebaum & Pollard's book:

I know how to show that the bottom one is the solution of the top equation, what I don't understand is why the interval on which the solution works is not:Differential equation:

Solution:

Give the interval on which the solution makes sense.

but instead:

as given the by author.

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Trying to understand this intuitively, I'm thinking that anything multiplied by zero is zero. So when , anything substituted on the left hand side gives zero. However, if is the solution, when , the RHS is always zero.

I don't see where the restriction comes from.

Thanks for your help!