Series Solutions - Examples & Applications

Conceptual Examples: Classification and Convergence

Classifying Ordinary and Singular Points

Determine and classify all singular points of the differential equation:
(x2−4)2y′′+(x−2)y′+xy=0(x^2 - 4)^2 y'' + (x - 2)y' + xy = 0

Classification Solution

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Radius of Convergence (Ratio Test)

Find the radius of convergence RR for the power series ∑n=1∞(−1)nn4nxn\sum_{n=1}^\infty \frac{(-1)^n n}{4^n} x^n.

Radius of Convergence Solution

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Power Series Method (Ordinary Points)

Power Series Problem (First-Order)

Find the power series solution centered at x0=0x_0 = 0 for:
y′−2xy=0y' - 2xy = 0

Power Series Solution

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Airy's Equation

Find the first few terms of the general series solution for Airy's Equation:
y′′−xy=0y'' - xy = 0

Airy's Equation Solution

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Method of Frobenius (Regular Singular Points)

Frobenius Method (Finding Indicial Roots)

Find the indicial equation and its roots for the differential equation:
2x2y′′−xy′+(1+x)y=02x^2 y'' - x y' + (1+x)y = 0

Indicial Equation Solution

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Frobenius Method (Full Solution)

Find the general solution of 2xy′′+y′+y=02x y'' + y' + y = 0 for x>0x > 0.

Full Frobenius Solution

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