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

By Power Series Method, the solution of the differential equation is

Let us look at some details.

By taking the derivative term by term,

Now, let us look at the differential equation.

by substituting the above power series in the equation,

by pulling the first term from the summation on the left,

by shifting the indices of the summation on the left by 2,

By matching coefficients,

and

Let us observe the odd terms.

Let us observe the even terms.

Hence,

#y=sum_{n=0}^infty c_0/{2^n cdot n!}x^{2n}=c_0 sum_{n=0}^infty{{x^{2n}}/{2^n}}/{n!} =c_0sum_{n=0}^infty{(x^2/2)^n}/{n!} =c_0e^{x^2/2}#

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To solve the differential equation ( y' = xy ) using power series, we express the solution as a power series in the form ( y(x) = \sum_{n=0}^{\infty} a_n x^n ). Then, we differentiate the series term by term to obtain an expression for ( y'(x) ). Substituting this expression into the original differential equation, we equate coefficients of like powers of ( x ) on both sides. By solving these equations recursively, we can determine the coefficients ( a_n ) for the power series solution.

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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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