what is meant by series solution of differential equation? why do we need series solution?

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what is meant by series solution of differential equation? why do we need series solution?

Mathematics
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At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

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A series solution to a differential equation is one of the form\[y=\sum_{n=0}^{\infty}a_nx^n\]assuming you have some differential equation where you're looking to solve for y(x). You then substitute the series into the differential equation with the aim of finding a recurrence relation for the coefficients. Series solutions are extremely useful since not all differential equations can be solved easily, or exactly, for some function...but the worlds of mathematics, science, engineering and economics all throw up differential equations that need to be solved quickly for use - this is where series solutions come into their own; when you don't know how to solve an equation, or have the time for insight, plug in a series solution and see what happens. It must be stipulated, though, that certain assumptions need to be met before applying series.

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