Second-Order Linear ODEs¶

Job: Solve constant- and variable-coefficient linear second-order equations; use reduction of order, variation of parameters, and Sturm–Liouville spectral structure.

Core¶

Standard form: $$ y''+p(x)y'+q(x)y=g(x),\qquad x\in I. $$ If $p,q,g$ continuous on $I$, IVPs at $x_0\in I$ have unique global solutions on $I$ (no finite-time blow-up for linear equations).

Homogeneous solution space is 2-dimensional. Wronskian of solutions $y_1,y_2$: $$ W=y_1 y_2'-y_2 y_1'=C\exp\Bigl(-\int p\,dx\Bigr)\quad\text{(Abel)}. $$ $W\not\equiv0$ iff $\{y_1,y_2\}$ is a fundamental set.

Constant coefficients $y''+by'+cy=0$: characteristic $r^2+br+c=0$.

Roots	Basis
$r_1\neq r_2$ real	$e^{r_1 x}, e^{r_2 x}$
$r$ repeated	$e^{rx}, x e^{rx}$
$\alpha\pm i\beta$	$e^{\alpha x}\cos\beta x$, $e^{\alpha x}\sin\beta x$

Nonhomogeneous: particular solution $y_p$ by undetermined coefficients (when $g$ is quasi-polynomial and annihilated by a constant-coeff operator) or variation of parameters: $$ y_p=u_1 y_1+u_2 y_2,\quad u_1'y_1+u_2'y_2=0,\quad u_1'y_1'+u_2'y_2'=g. $$

Reduction of order: one homogeneous solution $y_1\Rightarrow y_2=v y_1$ with $v'=\exp(-\int p)/y_1^2$.

Cauchy–Euler: $ax^2 y''+bxy'+cy=0$ — try $x^r$ on $(0,\infty)$; equivalent to constant-coeff after $x=e^t$.

Sturm–Liouville (regular): $$ -\bigl(p y'\bigr)'+q y=\lambda w y,\quad p>0,\ w>0\text{ on }[a,b], $$ self-adjoint BCs $\Rightarrow$ real simple eigenvalues $\lambda_n\uparrow\infty$, eigenfunctions orthogonal in $L^2(w)$, complete in $L^2(w)$. Oscillation: $n$th eigenfunction has $n$ zeros in $(a,b)$ (Sturm separation / comparison).

Green's function for nonhomogeneous BVP $Ly=g$ with homogeneous BCs: $y(x)=\int_a^b G(x,\xi)g(\xi)\,d\xi$, $G$ continuous, jump $1/p$ in $\partial_x G$ at $x=\xi$.

See it¶

Resonance: $y''+\omega^2 y=\cos\omega t$. Homogeneous basis $\{\cos\omega t,\sin\omega t\}$. Variation of parameters yields $y_p=\frac{t}{2\omega}\sin\omega t$ (secular growth). Fredholm: solvability of periodic problems requires $\int_0^{2\pi/\omega} g(t)\cdot(\text{kernel of }L^*)\,dt=0$.

Check¶

Why can't two solutions of $y''+p y'+q y=0$ both vanish at the same interior point unless they are linearly dependent?

If $y_1(x_0)=y_2(x_0)=0$, then $W(x_0)=0$. By Abel, $W\equiv0$ on the interval $\Rightarrow$ dependent. (Sturm separation is the refined global version.)

Sketch: how does one build $G(x,\xi)$ for $-y''=f$, $y(0)=y(1)=0$?

Solve homogeneous on $[0,\xi]$ and $[\xi,1]$ with BC at ends; match continuity of $G$ at $\xi$ and jump $G_x(\xi^+)-G_x(\xi^-)=-1$. Obtain $G(x,\xi)=\min(x,\xi)(1-\max(x,\xi))$.

Pitfalls¶

Complex roots: real form is preferred in applications; complex exponentials still valid over $\mathbb{C}$.
Undetermined coefficients fail for non-quasi-polynomial $g$; use variation of parameters.
Singular SL endpoints (Legendre, Bessel) need limit-point / limit-circle classification (Weyl).

Refs¶

Coddington–Levinson; Zettl, Sturm–Liouville Theory; Coddington, Theory of Ordinary Differential Equations.

Next¶

Foundations · Laplace transforms · PDE foundations · Fourier analysis

Roots	Basis
\(r_1\neq r_2\) real	\(e^{r_1 x}, e^{r_2 x}\)
\(r\) repeated	\(e^{rx}, x e^{rx}\)
\(\alpha\pm i\beta\)	\(e^{\alpha x}\cos\beta x\), \(e^{\alpha x}\sin\beta x\)