How do you find a Power Series solution of a partial differential equation?
By taking derivatives,
by shifting the indices of the first summation by 2,
by combining the summations,
Let us look at even coefficients.
Let us look at odd coefficients.
Hence, the solution can be written as:
by splitting into even terms and odd terms,
By recognizing the power series,
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To find a power series solution of a partial differential equation (PDE), follow these steps:
 Express the solution as a power series: Assume that the solution ( u(x, y) ) can be written as a power series in ( x ) and ( y ):
[ u(x, y) = \sum_{n=0}^\infty \sum_{m=0}^\infty a_{nm}x^n y^m ]

Substitute the power series into the PDE: Replace ( u(x, y) ) and its derivatives in the PDE with their corresponding power series expansions.

Equate coefficients of like powers of ( x ) and ( y ): Match coefficients of like powers of ( x ) and ( y ) on both sides of the PDE.

Solve the resulting system of algebraic equations: Solve the system of equations obtained by equating coefficients.

Determine the convergence of the solution: Analyze the convergence of the power series solution, considering the radius of convergence.

Verify the solution: Substitute the power series solution back into the original PDE to ensure it satisfies the equation.

Apply boundary or initial conditions: If boundary or initial conditions are given, apply them to determine the values of the coefficients in the power series solution.

Check for uniqueness: Ensure that the power series solution is unique within the given boundary or initial conditions.
By following these steps, you can find a power series solution of a partial differential equation.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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