The bookUnit 3 · Ordinary Differential Equations55 / 83

Linear ODE, Wronskian, variation of parameters, systems

Why this is asked: Wronskian non-zero ⇒ independent, but the converse needs the functions to solve a common linear ODE. For systems, the eigenvalues of A decide the growth rate of e^{At}.

Linear ODE, Wronskian and systems

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The Wronskian: all or nothing, by Abel's identityinteractivefree

Step 1 / 4Abel's identity

For $y^{''} + p (x) y^{'} + q (x) y = 0$ with $p, q$ continuous on an interval $I$ , the Wronskian of two solutions satisfies $W^{'} = - p W$ , so

$W (x) = W (x_{0}) exp (- \int_{x_{0}}^{x} p) .$

The exponential never vanishes.

The trap here

“W(f, g) ≡ 0 implies f and g are linearly dependent” — false

$f(x) = x^{2}, g(x) = x$ |x| on $\mathbb{R}$

The Wronskian vanishes identically but no constant multiple relates them. The implication holds only for solutions of a common linear ODE.