Distinguish which one is initial valued ordinary differential equation and boundary valued ordinary differential equation:
i) $y^{''} + 2 y = e^{x}, y (π) = 1; y^{'} (π) = 2$
i) $y^{''} + 2 y = e^{x}, y (0) = 1; y^{'} (1) = 1$

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Solution

Given the differential equations with the subsidiary conditions are
i) $y^{''} + 2 y = e^{x}, y (π) = 1; y^{'} (π) = 2$
ii) $y^{''} + 2 y = e^{x}, y (0) = 1; y^{'} (1) = 1$ .
Between them the first one is initial valued ordinary differential equation as the subsidiary condition is about the point $π$ .
And the second one is boundary valued as the subsidiary conditions are about the points $0$ and $1$ .

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