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For a second-order linear differential equationthe substitution $y=u\exp [-\frac{1}{2}\int p(x)dx]$ eliminates the first-derivative term and gives

Work on an interval where the change of variables is smooth and invertible, with $s=\dot{x}\ne 0$. If $v(t)=u(x(t))$, the chain rule gives $\dot{v}=s{u}_x$ and $\ddot{v}=s^2{u}_{xx}+\dot{s}{u}_x$. Eliminating $u_{xx}$ using the original second-order linear differential equation givesFor the real transformation as printed, take $s>0$. Set $v=s^{1/2}U$, which is the Liouville transformation of a second-order equation. Differentiating twice makes the coefficient of $\dot{U}$ cancel. Dividing by $s^{1/2}$ leavesThe additional term is exactly $s^{-1/2}[(\dot{s}/s)d(s^{1/2})/dt-d^2(s^{1/2})/d{t}^2]$, as required.

On an interval with $f>0$, choose $s=f^{-1/2}$ and therefore $t={\int }^x\sqrt{f(\zeta )}d\zeta$ up to a constant. Primes in the following expression mean derivatives with respect to $x$. We have $\dot{s}=-f^{\prime }/(2f^2)$ and $\ddot{s}=-f^{\prime \prime }/(2f^{5/2})+(f^{\prime }{)}^2/f^{7/2}$, giving the WKB correction termNeglecting $F$ gives $U=c_{+}{e}^t+c_{-}{e}^{-t}$. Since $u=s^{1/2}U$, the WKB approximation isChanging the lower integration limit only rescales the constants. Small $|F|$ is the stated local approximation criterion; rapid variation or a turning point can invalidate it, and small local residual alone need not imply uniform accuracy over an arbitrarily long interval.

The printed hypothesis only says $f\ne 0$. If real $f<0$ on the interval, the printed square roots require a consistent complex branch. Equivalently a real WKB approximation isThis supplies the oscillatory interpretation for the negative-$f$ case instead of silently assuming that the original nonzero hypothesis means positive. Smoothness through the derivatives used above and absence of zeros are required locally.

The characteristic polynomial of this second-order linear differential equation factors as $r^2-r-6=(r-3)(r+2)$. Its two distinct roots give the general solution $y=A{e}^{3x}+B{e}^{-2x}$. Boundedness as $x\to \infty$ forces $A=0$, and the initial condition gives $B=1$.

Divide the second-order difference equation by $h^2$ and set $x=nh$. The first quotient is a central finite difference approximation to $y^{\prime \prime }(x)$; the quotient $(y_{n+1}-y_{n-1})/(2h)$ approximates $y^{\prime }(x)$. Their Taylor series errors are $O(h^2)$ for a sufficiently smooth function. This is therefore a finite difference method for the second-order linear differential equation in part (a).

At a fixed $x\ge 0$, take $h\to 0$ with $n=x/h$ integral. Since $\log r_{-}=-2h+O(h^3)$, the exact bounded discrete solution satisfiesLikewise $(1-2h{)}^{x/h}\to e^{-2x}$. The decay selected by boundedness in the linear recurrence relation becomes exactly the decay selected by boundedness in the ordinary differential equation.

The defining Wronskian identity gives the first-order linear differential equationWork first on an interval where $y_1\ne 0$. Dividing by $y_1$ recognizes a derivative:This is reduction of order. Choose the nonzero Wronskian from the Abel identity to obtain linear independence. The constant $C$ only adds a multiple of $y_1$. If $y_1$ has zeros, this quotient formula is local; the resulting solution can be continued across ordinary zeros using the original second-order linear differential equation.

For the stated order of the two solutions, define their Wronskian byTaking a derivative cancels the two mixed products: $W^{\prime }=y_1{y}_2^{\prime \prime }-y_1^{\prime \prime }{y}_2$. Substituting the second-order linear differential equation for each second derivative cancels the $q$ terms and leavesThis is the Abel identity. On an interval with continuous coefficients, linear independence means $W(x_0)\ne 0$; hence the Wronskian never vanishes anywhere on that interval.

Classification and characteristic exponents. For a normalized second-order linear differential equation, an ordinary point at zero means $p$ and $q$ are real analytic functions there. A regular singular point is a singular point where $xp$ and $x^{2}q$ nevertheless extend as real analytic functions to zero. Write $p_0={\lim }_{x\to 0}xp(x)$ and $q_0={\lim }_{x\to 0}{x}^{2}q(x)$. The lowest-power coefficient in the Frobenius method gives the indicial equationIts roots are the characteristic exponents at a regular singular point. If their difference is not an integer, there are two linearly independent solutions $x^{{\sigma }_j}{A}_j(x)$, with $A_j$ real analytic and $A_j(0)\ne 0$. For distinct real roots ${\sigma }_1>{\sigma }_2$ with integer difference, a solution for the larger root exists, and a second has the form $x^{{\sigma }_2}B(x)+c{y}_1\log x$; the logarithmic coefficient can vanish. For a repeated root, the second solution has the form $y_1\log x+x^{{\sigma }_1}B(x)$ after normalization. This is a Logarithmic Frobenius solution. Complex conjugate roots $a\pm ib$ give real solutions whose leading behaviour is $x^a\cos (b\log x)$ and $x^a\sin (b\log x)$, with analytic-series corrections. All these descriptions are on $x>0$, or on a fixed complex branch.

For the present equation, division by $x$ gives $p=1+(1-m)/x$ and $q=(1-m)/x$. Therefore $xp=x+1-m$ and $x^{2}q=(1-m)x$ are real analytic. Zero is regular singular when $m\ne 1$; when $m=1$, it is ordinary after removing the common factor $x$. In that exceptional case the normalized equation is $y^{\prime \prime }+y^{\prime }=0$.

For the Frobenius method substitution, the lowest coefficient and the general linear recurrence relation areThese equations come respectively from the coefficients of $x^{\sigma -1}$ and $x^{n+\sigma -1}$. At $m=1$, the same formal exponents $0,1$ describe the two possible leading powers at an ordinary point, rather than singular behaviour.

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 $A{y}_1+B{y}_2$.