Wronskian

Consider the equation, where $P,Q$ are functions of $x$ alone, and which has $2$ fundamental solutions $u(x),v(x)$ :

y''+Py'+Qy=0

The Wronskian $w(x)$ of two solutions $u(x),v(x)$ of differential equation, is defined to be:

w(x) = \Bigg\lvert{ \begin{matrix} u(x) & v(x) \\ u'(x) & v'(x) \\ \end{matrix} } \Bigg\rvert

Theorem

Suppose $u$ and $v$ are two solutions of a 2nd order differential equation, in the form mentioned above.

$W(u,v)$ is always zero or never zero (in the intended range of solutions).

Proof

Consider the equation, where $P,Q$ are functions of $x$ alone.

y''+Py'+Qy=0

Let $u(x),v(x)$ be $2$ fundamental solutions of the equation:

u''+Pu'+Qu=0 \;\;\land\;\; v''+Pv'+Qv=0

w = \Bigg\lvert{ \begin{matrix} u & v \\ u' & v' \\ \end{matrix} } \Bigg\rvert =uv'-vu'

w'=uv''-vu''=-P[uv'-vu']=-Pw

By solving the above relation:

w=c\cdot\exp{\bigg(-\int{P}\,\text{d}x\bigg)}

Suppose there exists $x_0$ such that $w(x_0)=0$ . That implies $c=0$ . That implies $w$ is always $0$ .

Abel’s Forumla

The conclusion in the above proof is known as the Abel’s formula.

W(x)=c\exp{\bigg(-\int{P}\,\text{d}x\bigg)}

Theorem

\set{f_1, f_2,\dots, f_n}\;\text{are linearly dependent} \iff \forall x;\;W(f_1, f_2,\dots, f_n)(x) = 0 \;

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November 2, 2026 1 min read

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