Wronskian

Statement

Two solutions $X_1(t)$ and $X_2(t)$ of a homogeneous linear second-order ODE are linearly independent if and only if their Wronskian

$$W=\left|\begin{array}{cc}{X}_1(t_0)& X_2(t_0)\\ {\dot{X}}_1(t_0)& {\dot{X}}_2(t_0)\end{array}\right|=X_1(t_0){\dot{X}}_2(t_0)-{\dot{X}}_1(t_0)X_2(t_0)$$

is nonzero at some (equivalently, any) point $t_0$.

Assumptions

• The ODE $\ddot{x}+p(t)\dot{x}+q(t)x=0$ has continuous coefficients $p,q$ on an interval $I$.
• $X_1,X_2$ are solutions on $I$.

Proof Sketch

The general solution $x=c_1{X}_1+c_2{X}_2$ can be matched to arbitrary initial conditions $(x_0,u_0)$ by solving a $2\times 2$ linear system whose coefficient matrix has determinant $W$. By the invertibility criterion, a unique solution $(c_1,c_2)$ exists if and only if $W\ne 0$, which is equivalent to linear independence of $X_1$ and $X_2$.

Full Proof

Imposing $x(t_0)=x_0$ and $\dot{x}(t_0)=u_0$ on $x=c_1{X}_1+c_2{X}_2$ gives the system

$$c_1{X}_1(t_0)+c_2{X}_2(t_0)=x_0,c_1{\dot{X}}_1(t_0)+c_2{\dot{X}}_2(t_0)=u_0.$$

In matrix form:

$$\left(\begin{array}{cc}{X}_1(t_0)& X_2(t_0)\\ {\dot{X}}_1(t_0)& {\dot{X}}_2(t_0)\end{array}\right)\left(\begin{array}{c}{c}_1\\ c_2\end{array}\right)=\left(\begin{array}{c}{x}_0\\ u_0\end{array}\right).$$

This system has a unique solution for any $(x_0,u_0)$ if and only if the coefficient matrix is invertible, i.e.\ $W\ne 0$.

If $W=0$, then either $X_1$ and $X_2$ are linearly dependent (one is a scalar multiple of the other), or the pair fails to span the solution space, and initial conditions cannot always be satisfied uniquely. $\square$

Notes / Intuition

• $W\ne 0$ if and only if $X_1$ and $X_2$ are linearly independent as functions on $I$.
• The solution space of a second-order homogeneous linear ODE is two-dimensional; two independent solutions form a basis.
• Example: $X_1=\cos \omega t$, $X_2=\sin \omega t$ ($\omega \ne 0$) give $W=\omega \ne 0$ for all $t$, confirming independence.
• The Wronskian is used in homogeneous second-order ODEs to verify that the two solutions obtained from the characteristic roots are genuinely independent.

Links

• Homogeneous Second-Order ODE
• Principle of Superposition
• Characteristic Roots