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 1

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). $$

Note

If the Wronskian is non-zero at a point and zero on another point, $u,v$ cannot be solutions to the same differential equation. $$

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 2

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

Note

This theorem is valid only under some conditions, which are out of the scope of the course and not covered.