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Let $p(x)$ and $q(x)$ be two continuous functions on an interval $(a, b)$ and suppose

that $(y_1, y_2)$ and $(z_1, z_2)$ are two pairs of linearly independent solutions to the ODE

$y'' + p(x)y' + q(x)y = 0, x ∈ (a, b).$

Show that there exists a constant $c\ne0$ such that $W [y_1, y_2](x) = cW [z_1, z_2](x)$ for all $x ∈ (a, b)$.

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Abel's identity

This shows that the Wronskian can be computed somewhat independently of the solutions you pick simply by knowing the initial value which will be a constant multiple of the other.

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