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The standard Airy functions solve the Airy ordinary differential equation. Their Wronskian is $\mathrm{Ai}(Z){\mathrm{Bi}}^{\prime }(Z)-{\mathrm{Ai}}^{\prime }(Z)\mathrm{Bi}(Z)=1/\pi$, so they form a fundamental solution pair, including for complex $Z$.

For the Airy ordinary differential equation $y^{\prime \prime }=xy$, a power series obeys $a_2=0$ and $a_{m+3}=a_m/[(m+3)(m+2)]$. Setting its first two coefficients to $(1,0)$ or $(0,1)$ givesEmpty products are one. The ratio test proves convergence on the whole complex plane. Termwise differentiation verifies the equation, and the initial normalization gives Wronskian one. By the Abel identity, the Wronskian remains one everywhere, so these solutions span the whole solution space.

For $y_{tt}+e^{-2\epsilon t}(\epsilon y_t+y)=0$, $y(0)=0$, $y_t(0)=1$, set $r=e^{-2\epsilon t}/2$. The exact equation becomes $r{y}_{rr}+(1-r)y_r+y/(2{\epsilon }^2)=0$. Let $m(t)=M(-1/(2{\epsilon }^2),1,r)$, using the Kummer function. It is regular at $r=0$ and tends to one. The Wronskian $W=m{y}_t-m_{t}y$ obeys $W_t=-\epsilon e^{-2\epsilon t}W$, with $W(0)=M(-1/(2{\epsilon }^2),1,1/2)$. Integration gives the displayed exact late slope. The other local solution grows at most logarithmically in $r$, so $m_{t}y\to 0$ and $W\to y_t$. This provides a check on a WKB approximation even near phases where its leading predicted slope vanishes.

The inverse-scattering spectral problem associated with this KdV convention isFordifferentiation gives the Wronskian identity

A basis of the solution space of a homogeneous linear differential equation for the Mészáros equation on $y>0$ isSubstitution verifies both solutions; their Wronskian is $-1/[y\sqrt{1+y}]$, so they are independent. At small $y$, $D_{+}=1+O(y)$ and $D_{-}=\log (4/y)-3+O(y\log y)$. At large $y$, $D_{+}\sim 3y/2$ and $D_{-}\sim 4/(15y^{3/2})$. Matching the actual early-time radiation forcing determines which constant and logarithmic combination is excited.

Two bound-state solutions of the same one-dimensional time-independent Schrodinger equation have constant Wronskian. Their decay at infinity makes this constant zero, so the two solutions are linearly dependent. Every bound-state energy is therefore nondegenerate.