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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 nondimensional constant shear $U=y$ and streamwise wavenumber one, put $W=(D^2-1)\hat{v}$. The Orr-Sommerfeld equation gives $W^{\prime \prime }=[1+iRe(y-c)]W$. The displayed substitution transforms it into the Airy ordinary differential equation, because $(e^{i\pi /6}{)}^3=i$. The scale $R{e}^{-1/3}$ balances shear advection against viscous diffusion.

Assume the usual continuous-coefficient setting for a second-order linear differential equation. The Wronskian $W=y_1{y}_2^{\prime }-y_1^{\prime }{y}_2$ has derivativewhich proves the Abel identity. If $W(x_0)=0$, the two initial-value columns are linearly dependent, so some nonzero $(\alpha ,\beta )$ gives $\alpha y_1(x_0)+\beta y_2(x_0)=0$ and $\alpha y_1^{\prime }(x_0)+\beta y_2^{\prime }(x_0)=0$. The uniqueness theorem for ordinary differential equations makes that linear combination identically zero. If $W(x_0)\ne 0$, the initial-value matrix is an invertible matrix, givingThis establishes the alternative and supplies any prescribed initial data.

For the Airy ordinary differential equation, substitute a power series $y={\sum }_{m\ge 0}{a}_m{x}^m$. Coefficient comparison gives $a_2=0$ andChoose $(a_0,a_1)=(1,0)$ and $(0,1)$. The Airy power-series fundamental pair isEmpty products mean one. Their first terms are $y_1=1+x^3/6+x^6/180+\cdots$ and $y_2=x+x^4/12+x^7/504+\cdots$. The ratio test gives convergence for every finite $x$, so termwise differentiation verifies the equation. Their Wronskian is one at zero, and hence everywhere because $p=0$. Thus every solution is $A{y}_1+B{y}_2$.

Put $W=(D^2-1)\hat{v}$ and ${\delta }^3=R{e}^{-1}$. The constant-shear Orr-Sommerfeld equation givesLet $a=e^{i\pi /6}$ and $Z=a(y-c-i{\delta }^3)/\delta$. Since $d/dy=(a/\delta )d/dZ$ and $a^3=i$, the equation becomesThis is the Airy ordinary differential equation, yielding the Airy reduction of the Orr-Sommerfeld equation in constant shear. It is distinct from the evolution PDE represented by the existing Airy equation article.