Separation of variables

Theorem

Separation of variables

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Statement

Let \(L_x\) be a linear differential operator acting only on the variable \(x \in (a,b)\) and let \(L_t\) be a linear differential operator acting only on the variable \(t \in (c,d)\), and suppose \(u:(a,b)\times(c,d)\to\mathbb{R}\) solves the linear homogeneous PDE \(L_x u = L_t u\) (every term separated into an \(x\)-part and a \(t\)-part with no mixed derivatives and no term \(f(x,t)\) that fails to factor as \(g(x)h(t)\)). If \(u(x,t)=X(x)T(t)\) with \(X\not\equiv 0\) and \(T\not\equiv 0\), then there exists a constant \(\lambda\in\mathbb{C}\) (the separation constant) such that \(X\) solves the ODE eigenvalue problem \(L_xX=\lambda X\) on \((a,b)\) subject to the given linear homogeneous boundary conditions in \(x\), and \(T\) solves \(L_tT=\lambda T\) on \((c,d)\). Conversely, for each eigenvalue \(\lambda_n\) of the Sturm–Liouville problem for \(L_x\) with eigenfunction \(X_n\), and corresponding solution \(T_n\) of \(L_tT_n=\lambda_nT_n\), the product \(u_n(x,t)=X_n(x)T_n(t)\) solves the PDE and its boundary conditions in \(x\); if in addition \(\{X_n\}\) is a complete orthogonal system on \((a,b)\) (Sturm–Liouville theory), then the general solution satisfying a prescribed initial/data condition \(u(x,t_0)=\phi(x)\) is \(u(x,t)=\sum_n c_nX_n(x)T_n(t)\) with \(c_n\) the generalised Fourier coefficients of \(\phi\) with respect to \(\{X_n\}\).

Why it matters

Separation of variables is the mechanism by which the three canonical linear PDEs of mathematical physics — the heat equation, the wave equation, and Laplace's equation — are reduced from problems in infinitely many degrees of freedom to a countable family of ordinary differential equation problems. It converts a PDE into an eigenvalue problem for a self-adjoint operator, and the completeness of the resulting eigenfunctions (guaranteed by Sturm–Liouville theory) is exactly what licenses Fourier series, Fourier–Bessel series, and spherical harmonic expansions.

Its importance is not merely computational. The method reveals that the qualitative long-time or boundary behaviour of a linear evolution PDE is controlled by the spectrum \(\{\lambda_n\}\) of a single spatial operator: decay rates for the heat equation, resonant frequencies for the wave equation, and harmonic structure for Laplace's equation are all read off from this spectrum. It is the prototype for the spectral methods used throughout linear PDE theory and quantum mechanics.

Hypotheses

Linearity and homogeneity of the PDE.For the nonlinear PDE \(u_t = u\,u_{xx}\), substituting \(u=XT\) gives \(XT' = XT\,X''T^2\), i.e. \(T'/T^2 = X''\), which does not separate into a function of \(x\) alone equal to a function of \(t\) alone in a way that yields two independent linear ODEs — the ansatz fails to decouple. Even for a linear but inhomogeneous PDE such as \(u_t = u_{xx} + f(x,t)\) with \(f\) not a product \(g(x)h(t)\), the substitution \(u=XT\) does not eliminate both variables from a single equation, and the method must be replaced by eigenfunction expansion with time-dependent coefficients. The operator splits additively with no mixed partial derivatives across the separated variables.For \(u_{xt}+u=0\), writing \(u=XT\) gives \(X'T'+XT=0\), i.e. \(X'/X = -T/T' \) is not achievable as "function of \(x\) = function of \(t\)" without dividing by \(X'T'\), and one obtains \((X'/X)(T'/T)=-1\), a genuinely coupled relation that does not force each side to be a separate constant — the classical separation argument breaks down for cross-derivative terms. The spatial boundary conditions are linear and homogeneous.If instead \(u(a,t)=1\) (an inhomogeneous Dirichlet condition) for all \(t\), then a product solution \(X(x)T(t)\) forces \(X(a)T(t)=1\) for all \(t\), which is impossible unless \(T\) is constant — a single product ansatz cannot carry a nonzero, non-constant boundary datum. One must first subtract off a particular solution (e.g. a steady-state) to homogenise the boundary conditions before separating. Completeness of the eigenfunction system \(\{X_n\}\) (for the superposition step).Without completeness — e.g. if the boundary conditions do not render \(L_x\) self-adjoint (non-Sturm–Liouville boundary conditions, such as a mismatched Robin condition with the wrong sign making the operator non-symmetric) — the family \(\{X_n\}\) may fail to span the relevant function space, so an arbitrary initial datum \(\phi(x)\) cannot be represented as \(\sum c_nX_n(x)\), and the series solution fails to satisfy the initial condition.

Proof

u(x,t)=X(x)T(t),\quad X\not\equiv 0,\ T\not\equiv 0

Ansatz: seek non-trivial product solutions. This is a restriction of the solution space, not yet a claim about all solutions of the PDE; justified retroactively by completeness in Step 6. A

L_x u = L_t u \;\Longrightarrow\; T(t)\,(L_xX)(x) = X(x)\,(L_tT)(t)

Substitute the ansatz into the PDE. Since \(L_x\) acts only on \(x\) and \(T\) is independent of \(x\), \(L_x(XT)=T\cdot L_xX\); symmetrically \(L_t(XT)=X\cdot L_tT\). This uses only linearity of \(L_x,L_t\) and that partial differentiation in \(x\) treats \(T(t)\) as a constant factor (and vice versa). A

\text{Let } S=\{x\in(a,b): X(x)\neq 0\},\quad R=\{t\in(c,d): T(t)\neq 0\}.\ S,R \text{ are open and dense (real-analytic or } C^1\text{ solutions of an ODE are not identically zero on any subinterval unless globally zero).}

Since \(X\not\equiv0\) and \(X\) solves a linear ODE (once \(\lambda\) is fixed) it is real-analytic on \((a,b)\) wherever the ODE has analytic coefficients, so its zero set is discrete; hence \(S\) is dense open, and similarly \(R\). This uses the identity theorem for solutions of linear ODEs with analytic coefficients. C †

\text{For } (x,t)\in S\times R:\quad \frac{(L_xX)(x)}{X(x)} = \frac{(L_tT)(t)}{T(t)}

Divide both sides of Step 2 by \(X(x)T(t)\neq 0\) on \(S\times R\). This is legal exactly where neither factor vanishes. A

\frac{(L_xX)(x)}{X(x)} = \lambda \text{ for all } x\in S, \qquad \frac{(L_tT)(t)}{T(t)} = \lambda \text{ for all } t\in R, \qquad \lambda \in \mathbb{C} \text{ constant.}

The left side of Step 4 depends only on \(x\), the right only on \(t\); fixing any \(t_0\in R\) shows the left side equals the constant \((L_tT)(t_0)/T(t_0)\) for every \(x\in S\), and symmetrically. Hence both sides equal one and the same constant \(\lambda\), independent of \(x\) and \(t\). This is the crux "separation" step: a function of \(x\) alone that equals a function of \(t\) alone, on a set with nonempty interior in each variable, must be constant. B

L_xX=\lambda X \text{ on } (a,b), \qquad L_tT=\lambda T \text{ on } (c,d)

By continuity, the identities of Step 5 extend from the dense set \(S\) (resp. \(R\)) to the whole interval \((a,b)\) (resp. \((c,d)\)), since \(L_xX\), \(X\), \(L_tT\), \(T\) are continuous and \(S,R\) are dense. The boundary conditions on \(u\) are linear and homogeneous by hypothesis, so they pass to conditions purely on \(X\) (e.g. \(u(a,t)=0\;\forall t\in R\) with \(T\not\equiv0\) forces \(X(a)=0\)). A

\{(\lambda_n,X_n)\}_{n\in\mathbb{N}} = \text{eigenpairs of the Sturm–Liouville problem } L_xX=\lambda X,\ \text{(homog. BCs)}

Named lemma: Sturm–Liouville theory guarantees that a regular (or suitably singular) self-adjoint Sturm–Liouville problem has a countable, discrete, real spectrum \(\lambda_1\lt\lambda_2\lt\cdots\to\infty\) (or \(-\infty\), depending on sign convention) with eigenfunctions \(X_n\) forming a complete orthogonal system in \(L^2(a,b;w)\) for the appropriate weight \(w\). Cited, not reproved here. C ‡

u_n(x,t)=X_n(x)T_n(t) \text{ solves } L_xu=L_tu \text{ and the } x\text{-boundary conditions, for each } n

Direct substitution using linearity of \(L_x,L_t\) as in Step 2, run in reverse, with \(\lambda=\lambda_n\). Each \(u_n\) is an exact particular solution. A

u(x,t) = \sum_{n=1}^{\infty} c_n X_n(x) T_n(t), \qquad c_n = \frac{\int_a^b \phi(x)X_n(x)\,w(x)\,dx}{\int_a^b X_n(x)^2\,w(x)\,dx}

By linearity of the PDE and its boundary conditions, any finite (and, subject to convergence, infinite) linear combination of the \(u_n\) is again a solution (superposition principle). Completeness of \(\{X_n\}\) from Step 7 guarantees \(\phi(x)=\sum c_nX_n(x)\) in \(L^2\) (and pointwise under standard smoothness/Dirichlet-kernel conditions), matching the data \(u(x,t_0)=\phi(x)\) after choosing \(T_n(t_0)\) normalised to \(1\) (or absorbing that constant into \(c_n\)). B §

Result

L_xu=L_tu,\ u=XT \ \Longrightarrow\ L_xX=\lambda X,\ \ L_tT=\lambda T \qquad u(x,t)=\sum_n c_nX_n(x)T_n(t)

Reading. A linear, homogeneous PDE in which the operators for each variable act independently reduces, on the product ansatz, to two ODE eigenvalue problems sharing the same constant \(\lambda\); the spatial one, being self-adjoint under homogeneous boundary conditions, has a complete set of eigenfunctions, and every admissible solution is a (generalised) Fourier series built from these eigenfunctions.

Scope. Applies to linear, homogeneous, constant- (or separably-varying-) coefficient PDEs on product domains with linear homogeneous boundary conditions reducible to a regular or singular Sturm–Liouville problem in the separated variable — covers the heat, wave, and Laplace equations on intervals, rectangles, disks, cylinders, and spheres under Dirichlet/Neumann/Robin/periodic conditions. It does not directly apply to nonlinear PDEs, PDEs with genuinely mixed-derivative or non-factoring source terms, or inhomogeneous boundary data (without prior homogenisation).

Corollaries & converses

If the PDE and boundary conditions are as above and \(\phi\in L^2(a,b;w)\), the series solution converges in \(L^2\); under additional smoothness of \(\phi\) and compatibility with the boundary conditions, convergence is uniform (Sturm–Liouville completeness plus standard Fourier-series regularity results).
For the heat equation \(u_t=ku_{xx}\), every eigenvalue \(\lambda_n\le 0\) forces \(T_n(t)=e^{\lambda_n kt}\) to decay, giving the smoothing/decay property of parabolic separated solutions; for the wave equation \(u_{tt}=c^2u_{xx}\), \(\lambda_n\lt0\) forces \(T_n\) oscillatory, giving normal modes.
Converse: not every solution of a separable linear homogeneous PDE with homogeneous BCs is itself a single product \(X(x)T(t)\) — only the eigenfunction expansion (an infinite sum of products, not one product) recovers the general solution. Uniqueness of \(\lambda\) per product mode does not imply the full solution set is one-dimensional.
If the boundary conditions are inhomogeneous, the theorem does not apply directly; the converse fails outright, as shown in Hypotheses above — a homogenising particular solution must be subtracted first.

Fails without

Drop linearity: the nonlinear PDE \(u_t + uu_x = 0\) (inviscid Burgers) admits no consistent product ansatz: substituting \(u=XT\) gives \(XT' + XX'T^2 = 0\), i.e. \(T'/T^2 = -X'/X\), which is not of the "function of \(t\)" = "function of \(x\)" form needed to force a constant, because the left side still carries a hidden \(t\)-dependence through the nonlinear coupling of \(T\) with itself; more directly, no nontrivial choice of \(X,T\) solves the PDE for arbitrary initial data, and the true solution generally develops shocks that no product form can represent.
Drop homogeneous boundary conditions: for \(u_t=u_{xx}\) on \((0,\pi)\) with \(u(0,t)=1\), \(u(\pi,t)=0\), a product solution \(X(x)T(t)\) would need \(X(0)T(t)=1\) for all \(t\); since \(T\) is not identically constant (it must decay/vary to match the PDE in general), no single product satisfies this. The correct method subtracts the steady-state \(v(x)=1-x/\pi\) (solving \(v''=0\) with the same inhomogeneous data) and separates variables only for \(w=u-v\), which does satisfy homogeneous boundary conditions.
Drop the "no mixed derivative" structure: the transport-diffusion-cross equation \(u_{xt}=u\) does not separate: substituting \(u=XT\) gives \(X'T'=XT\), i.e. \((X'/X)(T'/T)=1\), a product (not sum) of purely-\(x\) and purely-\(t\) quantities equalling a constant, which does not force each factor individually to be constant (e.g. \(X'/X=g(x)\), \(T'/T=1/g(x)\) can vary with an auxiliary free function), so the reduction to two decoupled ODEs fails.

Common errors

Assuming the separation constant \(\lambda\) must be real and negative (or positive) by fiat, rather than determining its admissible sign/values from the boundary conditions — sign errors here silently discard the only nontrivial (eigenfunction) solutions.
Forgetting to check \(X\equiv0\) or \(T\equiv0\) does not count as a valid separated solution, then wrongly concluding "no solutions exist" when the trivial case is excluded improperly, or conversely including trivial solutions in the eigenfunction sum.
Applying separation of variables directly to a PDE with inhomogeneous boundary data without first homogenising by subtracting a particular/steady-state solution.
Treating the finite sum obtained after separating one or two terms as the general solution, instead of summing over the full countable eigenvalue spectrum guaranteed by Sturm–Liouville theory.
Confusing the initial-time coefficient formula \(c_n=\langle \phi,X_n\rangle/\langle X_n,X_n\rangle\) with an unweighted inner product when the Sturm–Liouville problem carries a nontrivial weight \(w(x)\) (e.g. Bessel or Legendre problems), producing wrong Fourier coefficients.

Discussion

Separation of variables dates, in essence, to Daniel Bernoulli's 1753 treatment of the vibrating string via superposed sinusoidal modes, and to Fourier's 1822 Théorie analytique de la chaleur, which used it systematically for the heat equation and thereby launched Fourier analysis. What began as a computational trick for specific equations was only later understood, through Sturm and Liouville's work in the 1830s, as an instance of spectral theory for self-adjoint operators — the separated ODE eigenvalue problem is precisely a Sturm–Liouville problem, and its completeness theorem is what makes the method rigorous rather than merely formal.

The method generalises naturally to more than two variables and to curved domains: separating Laplace's equation in spherical coordinates yields spherical harmonics and the associated Legendre equation; in cylindrical coordinates it yields Bessel functions. In each case the "constant" \(\lambda\) from one separation becomes a variable input to the next separation, producing a chain of eigenvalue problems — this is the structural origin of the special functions of mathematical physics.

From a more modern operator-theoretic viewpoint, separation of variables is the statement that if \(H=H_1\otimes H_2\) (a tensor product of Hilbert spaces) and an operator has the form \(A\otimes I + I\otimes B\) with \(A,B\) self-adjoint, then its eigenfunctions can be taken to be simple tensors \(\psi_n\otimes\varphi_m\) of eigenfunctions of \(A\) and \(B\) separately, with eigenvalues \(\alpha_n+\beta_m\); the "separation constant" \(\lambda\) is exactly an eigenvalue of \(A\) (equivalently \(-B\), depending on sign convention). This is the same principle underlying the treatment of multi-particle Schrödinger operators without interaction terms.

Common misconception: that separation of variables proves the PDE has product-form solutions only. In fact the theorem's full content is the converse-superposition step (Step 9): the general solution is an infinite linear combination of product solutions, and it is generically not itself a single product — this is frequently glossed over in first exposure, leaving students unable to explain why the method still produces the "general" solution.

Worked examples

\text{Heat equation: } u_t = k\,u_{xx},\ \ 0\lt x\lt L,\ t\gt 0,\quad u(0,t)=u(L,t)=0,\quad u(x,0)=\phi(x)

Homogeneous linear PDE, homogeneous Dirichlet BCs — hypotheses of the theorem are met with \(L_xX=X''\), \(L_tT=T'/k\). A

X''=\lambda X,\ X(0)=X(L)=0 \quad\Longrightarrow\quad \lambda_n=-\left(\frac{n\pi}{L}\right)^2,\ \ X_n(x)=\sin\!\left(\frac{n\pi x}{L}\right),\ n=1,2,\dots

Standard regular Sturm–Liouville eigenvalue problem; only \(\lambda\lt0\) gives nontrivial solutions satisfying both endpoint conditions (checking \(\lambda=0\) and \(\lambda\gt0\) gives only \(X\equiv0\)). B

T_n' = k\lambda_n T_n \ \Longrightarrow\ T_n(t) = \exp\!\left(-k\left(\frac{n\pi}{L}\right)^2 t\right)

Solve the companion first-order linear ODE for \(T\) with the same \(\lambda_n\) from Step 2. A

u(x,t)=\sum_{n=1}^\infty b_n\sin\!\left(\frac{n\pi x}{L}\right)e^{-k(n\pi/L)^2t}, \qquad b_n=\frac{2}{L}\int_0^L \phi(x)\sin\!\left(\frac{n\pi x}{L}\right)dx

Superpose (Step 9 of the proof) and fix \(b_n\) by matching \(t=0\) via orthogonality of \(\{\sin(n\pi x/L)\}\) on \((0,L)\) with weight \(1\), a special case of Sturm–Liouville completeness. B

u(x,t)=\sum_{n=1}^{\infty} b_n \sin\!\left(\frac{n\pi x}{L}\right) e^{-k(n\pi/L)^2 t}

Reading. The solution to the Dirichlet heat problem is the Fourier sine series of the initial data, each mode damped exponentially at a rate growing with \(n^2\).

Scope. Valid for any \(\phi\in L^2(0,L)\); higher modes decay fastest, so for \(t\gt0\) the solution instantly becomes smooth even for rough \(\phi\).

\text{Laplace's equation on a disk: } u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}u_{\theta\theta}=0,\ \ 0\le r\lt R,\ \ u(R,\theta)=f(\theta)

Linear, homogeneous PDE in \((r,\theta)\); write \(u=X(r)T(\theta)\) but note here the operator is not literally additive in \(x,t\) form — rewrite \(r^2u_{rr}+ru_r=-u_{\theta\theta}\) so that each side depends on a single variable after dividing by \(u\), matching the theorem's hypothesis with \(L_r X = r^2X''+rX'\) and \(L_\theta T = -T''\), coupled by \(\lambda\) via \(r^2X''+rX' = \lambda X\), \(T''=-\lambda T\). A

T''=-\lambda T,\quad T(\theta+2\pi)=T(\theta)\ \Longrightarrow\ \lambda_n=n^2,\ \ T_n(\theta)=\cos n\theta,\ \sin n\theta,\ n=0,1,2,\dots

Periodic boundary conditions in \(\theta\) (physical single-valuedness of \(u\)) form a regular Sturm–Liouville problem on a circle; eigenvalues are the nonnegative integers squared, with two independent eigenfunctions per \(n\ge1\). B

r^2X''+rX'-n^2X=0 \ \Longrightarrow\ X_n(r)=r^{n} \ (\text{rejecting } r^{-n}\text{, singular at } r=0)

Euler-type ODE for \(X\), companion to \(\lambda_n=n^2\); boundedness of \(u\) at the origin (a physical regularity condition, playing the role of a boundary condition at the singular endpoint \(r=0\)) eliminates the second, singular solution \(r^{-n}\). B

u(r,\theta)=\frac{a_0}{2}+\sum_{n=1}^{\infty} r^n\left(a_n\cos n\theta + b_n\sin n\theta\right),\qquad a_n=\frac{1}{\pi R^n}\int_0^{2\pi} f(\theta)\cos n\theta\,d\theta,\ \ b_n=\frac{1}{\pi R^n}\int_0^{2\pi} f(\theta)\sin n\theta\,d\theta

Superpose all product modes and match at \(r=R\) using orthogonality/completeness of \(\{\cos n\theta,\sin n\theta\}\) on \((0,2\pi)\), the Sturm–Liouville eigenbasis from Step 2. B

u(r,\theta)=\frac{a_0}{2}+\sum_{n=1}^{\infty} r^n(a_n\cos n\theta+b_n\sin n\theta)

Reading. Harmonic functions on a disk, bounded at the centre, are exactly power series in \(r\) with Fourier-in-\(\theta\) coefficients read off from the boundary data \(f\).

Scope. Valid for \(f\in L^2(0,2\pi)\); this is the Poisson-kernel representation of the Dirichlet problem on the disk in disguise.

Problems

Solve \(u_t=ku_{xx}\) on \(0\lt x\lt L\) with Neumann conditions \(u_x(0,t)=u_x(L,t)=0\) and \(u(x,0)=\phi(x)\).
Solution
Separating gives \(X''=\lambda X\), \(X'(0)=X'(L)=0\). Testing signs: \(\lambda=0\) gives \(X_0=1\) (constant, admissible); \(\lambda=-(n\pi/L)^2\) gives \(X_n(x)=\cos(n\pi x/L)\), \(n\ge1\) (checking \(X_n'(0)=0\) automatically, \(X_n'(L)=-\frac{n\pi}{L}\sin(n\pi)=0\) automatically). \(\lambda\gt0\) gives only the trivial solution. Then \(T_n(t)=e^{-k(n\pi/L)^2t}\) (with \(T_0\equiv\) constant). General solution: \(u(x,t)=\frac{a_0}{2}+\sum_{n\ge1}a_n\cos(n\pi x/L)e^{-k(n\pi/L)^2t}\), \(a_n=\frac{2}{L}\int_0^L\phi(x)\cos(n\pi x/L)dx\). As \(t\to\infty\), \(u\to a_0/2\), the average of \(\phi\), reflecting conservation of total heat under insulated (Neumann) ends.
Explain, using the proof above, why separation of variables cannot be applied directly to \(u_t = u_{xx} + \sin x\) without modification, and show how to fix it.
Solution
The PDE is linear but inhomogeneous: writing it as \(L_xu - L_tu = -\sin x\) with \(L_xu=u_{xx}\), \(L_tu=u_t\), the right-hand side is not zero, so Step 2 of the proof (which relies on \(L_xu=L_tu\) with no extra source term) does not apply — substituting \(u=XT\) gives \(XT'=X''T+\sin x\), and dividing by \(XT\) leaves an extra term \(\sin x/(XT)\) that still mixes \(x\) and \(t\), so the two sides cannot be equated to a single constant. Fix: find any particular steady solution \(v(x)\) with \(v''=-\sin x\), e.g. \(v(x)=\sin x\) works since \(v''=-\sin x\); actually solving \(v''+\sin x=0\Rightarrow v''=-\sin x\Rightarrow v=\sin x\) (check: \(v''=-\sin x\), matches). Let \(w=u-v\); then \(w_t=w_{xx}\), homogeneous, and separation applies to \(w\) with whatever homogeneous boundary conditions \(w\) inherits (assuming the original BCs on \(u\) match \(v\)'s boundary values, else adjust \(v\)).
The wave equation \(u_{tt}=c^2u_{xx}\) on \(0\lt x\lt L\) with \(u(0,t)=u(L,t)=0\) is separated as \(u=X(x)T(t)\). Derive the ODEs for \(X\) and \(T\), find the eigenvalues, and give the general (superposed) solution given \(u(x,0)=\phi(x)\), \(u_t(x,0)=\psi(x)\).
Solution
Substituting: \(XT''=c^2X''T\Rightarrow T''/(c^2T)=X''/X=\lambda\) (constant, by Step 5 of the proof). Boundary conditions force \(X(0)=X(L)=0\), giving \(\lambda_n=-(n\pi/L)^2\), \(X_n=\sin(n\pi x/L)\), as in Example 1. Then \(T_n''=-c^2(n\pi/L)^2T_n\), so \(T_n(t)=A_n\cos(cn\pi t/L)+B_n\sin(cn\pi t/L)\). Superposing: \(u(x,t)=\sum_{n\ge1}\sin(n\pi x/L)\left[A_n\cos(cn\pi t/L)+B_n\sin(cn\pi t/L)\right]\). Matching \(t=0\): \(A_n=\frac{2}{L}\int_0^L\phi(x)\sin(n\pi x/L)dx\) (Fourier sine coefficient of \(\phi\)); differentiating in \(t\) and setting \(t=0\): \(B_n\cdot\frac{cn\pi}{L}=\frac{2}{L}\int_0^L\psi(x)\sin(n\pi x/L)dx\), so \(B_n=\frac{2}{cn\pi}\int_0^L\psi(x)\sin(n\pi x/L)dx\).
Show directly (without invoking the theorem) that if \(X''/X=\lambda_1\) on a dense open set and also \(X''/X=\lambda_2\) on the same set, with \(\lambda_1\ne\lambda_2\), then \(X\equiv0\) there — confirming the "must be constant" claim of Step 5 is not vacuous.
Solution
If \(X''/X=\lambda_1\) and \(X''/X=\lambda_2\) hold simultaneously at a point where \(X\ne0\), then \(\lambda_1=\lambda_2\), contradicting \(\lambda_1\ne\lambda_2\). Hence at every point of the dense set, \(X=0\) is forced — but \(X\) continuous and zero on a dense set implies \(X\equiv0\) everywhere by continuity. This confirms that Step 5's separation argument is a genuine constraint (a nonzero \(X\) pins down a single, well-defined \(\lambda\)), not an empty tautology.
A student separates \(u_t=u_{xx}\), \(0\lt x\lt \pi\), \(u(0,t)=0\), \(u(\pi,t)=0\), and reports the general solution as \(u(x,t)=\sin(x)e^{-t}\) alone, having found "the" eigenvalue \(\lambda=-1\) and stopped. Identify the error and correct it, given \(u(x,0)=\phi(x)=x(\pi-x)\).
Solution
Error: the Sturm–Liouville problem \(X''=\lambda X\), \(X(0)=X(\pi)=0\) has infinitely many eigenvalues \(\lambda_n=-n^2\), \(n=1,2,3,\dots\), not just \(n=1\); the student stopped after finding one mode instead of summing over the full countable spectrum (a "common error" flagged above). Correct solution: \(u(x,t)=\sum_{n=1}^\infty b_n\sin(nx)e^{-n^2t}\), \(b_n=\frac{2}{\pi}\int_0^\pi x(\pi-x)\sin(nx)\,dx\). Computing, \(b_n=\frac{2}{\pi}\cdot\frac{4}{n^3}\) for odd \(n\) and \(0\) for even \(n\) (standard integral, using integration by parts twice and \(\cos(n\pi)=(-1)^n\)), i.e. \(b_n=\frac{8}{\pi n^3}\) for \(n\) odd, \(0\) for \(n\) even. So \(u(x,t)=\frac{8}{\pi}\sum_{k=0}^\infty \frac{\sin((2k+1)x)}{(2k+1)^3}e^{-(2k+1)^2t}\).