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Part CCSIR NET December 2023

Existence–uniqueness, Picard, Lipschitz

Consider the initial value problem $x^{2} y^{''} - 2 x^{2} y^{'} + (4 x - 2) y = 0, y (0) = 0$ . Suppose $y = φ (x)$ is a polynomial solution satisfying $φ (1) = 1$ . Which of the following statements are true?

A. $φ (4) = 16$ ✓
B. $φ (2) = 2$
C. $φ (5) = 25$ ✓
D. $φ (3) = 3$

Solution

Try $y = x^{2} : 2 x^{2} - 4 x^{3} + 4 x^{3} - 2 x^{2} = 0$ . So $φ (x) = x^{2}$ , giving $φ (4) = 16$ and $φ (5) = 25$ .

Related counterexample: Every IVP has a unique solution

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From Ordinary Differential Equations › Existence–uniqueness, Picard, Lipschitz

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