# How do you use Power Series to solve the differential equation #y''=y# ?

By matching each coefficients, #(n+2)(n+1)c_{n+2}=c_n Rightarrow c_{n+2}=c_n/{(n+2)(n+1)}#

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To solve the differential equation (y'' = y) using power series, you assume a power series solution of the form (y(x) = \sum_{n=0}^{\infty} a_n x^n). Then, you differentiate this expression twice, substitute it into the given differential equation, and equate coefficients of like powers of (x). This process generates a recurrence relation for the coefficients (a_n), which you can solve to find their values. Finally, you substitute these values back into the power series solution to obtain the solution to the differential equation.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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