Ordinary Differential Equations

An ordinary differential equation relates an unknown function of one independent variable to its derivatives. In quantum mechanics, ODEs describe time evolution in finite systems, one-dimensional stationary states, radial and angular factors after separation of variables, and many approximation schemes.

The differential equation alone rarely defines the whole problem. One must also specify an interval, regularity assumptions, initial or boundary data, and, for eigenvalue problems, the parameter values to be determined.

Order, Linearity, and Normal Form

An $n$th-order ODE can be written schematically as

$$F\left( x,y,y',\ldots,y^{(n)} \right)=0.$$

It is linear when it has the form

$$\begin{aligned} a_n(x)y^{(n)} &+a_{n-1}(x)y^{(n-1)} \\ &+\cdots+a_0(x)y=g(x). \end{aligned}$$

Where $a_n(x)\ne0$, division by the leading coefficient gives normal form:

$$\begin{aligned} y^{(n)} &+p_{n-1}(x)y^{(n-1)} \\ &+\cdots+p_0(x)y=r(x). \end{aligned}$$

The associated homogeneous equation has $r=0$. If $y_p$ is one particular solution of the inhomogeneous equation and $y_1,\ldots,y_n$ form a basis of homogeneous solutions, then every solution is

$$y=y_p+\sum_{j=1}^{n}c_jy_j.$$

The superposition principle applies to homogeneous linear equations. It does not apply to nonlinear equations such as $y'=y^2$.

First-Order Initial-Value Problems

A first-order initial-value problem has the form

$$\begin{aligned} y'&=f(x,y),\\ y(x_0)&=y_0. \end{aligned}$$

The Picard–Lindelöf theorem gives a local existence-and-uniqueness criterion. If $f$ is continuous near $(x_0,y_0)$ and locally Lipschitz in $y$, then there is an interval around $x_0$ on which exactly one solution satisfies the initial condition.

A convenient sufficient condition is continuity of both $f$ and $\partial f/\partial y$ in a rectangle around the initial point. These hypotheses are sufficient, not necessary.

Existence and uniqueness are distinct questions:

• continuity of $f$ alone can give local existence without uniqueness;
• a locally unique solution may still cease to exist at a finite endpoint;
• singular coefficients can prevent the theorem from applying.

For example,

$$y'=y^2, \qquad y(0)=1$$

has the unique local solution

$$y(x)=\frac{1}{1-x},$$

but it blows up at $x=1$. A local theorem does not imply global existence.

Nonuniqueness When Lipschitz Control Fails

Consider

$$y'=2\sqrt{\lvert y\rvert}, \qquad y(0)=0.$$

The right side is continuous but not locally Lipschitz at $y=0$. One solution is $y(x)=0$. For every $a\ge0$, another solution on $x\ge0$ is

$$y_a(x) = \begin{cases} 0,&0\le x\le a,\\ (x-a)^2,&x\ge a. \end{cases}$$

Each solution waits at zero for an arbitrary time and then departs. This example shows why checking only continuity is not enough when a unique evolution is required.

First-Order Linear Equations

A linear first-order equation is

$$y'+a(x)y=b(x).$$

Define an integrating factor

$$\mu(x) = \exp\left( \int^x a(s)\,ds \right).$$

Then

$$(\mu y)'=\mu b,$$

so

$$y(x) = \frac{1}{\mu(x)} \left[ C+\int^x\mu(s)b(s)\,ds \right].$$

The constant $C$ is fixed by one initial condition. Changing the lower limits only changes the way $C$ is written.

As an example,

$$y'+2xy=2x$$

has $\mu=e^{x^2}$ and

$$y(x)=1+Ce^{-x^2}.$$

Linear Second-Order Equations

The standard normalized form is

$$y''+p(x)y'+q(x)y=r(x).$$

If $p,q,r$ are continuous on an interval $I$, then specifying

$$y(x_0)=y_0, \qquad y'(x_0)=v_0$$

at $x_0\in I$ determines a unique solution throughout the interval on which the coefficients remain regular.

The homogeneous solution space is two-dimensional. Two solutions $y_1,y_2$ form a fundamental pair when they are linearly independent. Every homogeneous solution is then

$$y_h=c_1y_1+c_2y_2.$$

Initial data determine $c_1,c_2$ through a two-by-two linear system.

Wronskian and Abel’s Identity

The Wronskian of two differentiable functions is

$$W[y_1,y_2](x) = \begin{vmatrix} y_1&y_2\\ y_1'&y_2' \end{vmatrix} =y_1y_2'-y_1'y_2.$$

For two solutions of

$$y''+p(x)y'+q(x)y=0,$$

differentiation and substitution give

$$W'=-pW.$$

Hence Abel’s identity is

$$W(x) = W(x_0) \exp\left( -\int_{x_0}^{x}p(s)\,ds \right).$$

If the Wronskian is nonzero at one point, it is nonzero throughout the regular interval and the solutions are linearly independent. If it vanishes at one point, it vanishes everywhere on that interval.

The Wronskian test relies on both functions solving the same linear equation. For arbitrary differentiable functions, a Wronskian that vanishes identically does not always imply linear dependence without extra assumptions.

Constant-Coefficient Equations

For

$$y''+ay'+by=0,$$

try $y=e^{rx}$. The characteristic polynomial is

$$r^2+ar+b=0.$$

The root structure determines a fundamental pair:

• distinct roots $r_1,r_2$ give $e^{r_1x}$ and $e^{r_2x}$;
• a repeated root $r$ gives $e^{rx}$ and $xe^{rx}$;
• roots $\alpha\pm i\beta$ give the real pair $e^{\alpha x}\cos\beta x$ and $e^{\alpha x}\sin\beta x$.

For the oscillatory equation

$$y''+k^2y=0, \qquad k>0,$$

the general solution is

$$y=A\cos(kx)+B\sin(kx).$$

The differential equation allows every $k$. Boundary data may restrict $k$ to a discrete set.

Variation of Parameters

Let $y_1,y_2$ be a fundamental pair for the homogeneous equation

$$y''+p(x)y'+q(x)y=0.$$

A particular solution of

$$y''+p(x)y'+q(x)y=r(x)$$

can be written

$$\begin{aligned} y_p(x) &= -y_1(x) \int_{x_0}^{x} \frac{y_2(s)r(s)}{W(s)}\,ds\\ &\quad+ y_2(x) \int_{x_0}^{x} \frac{y_1(s)r(s)}{W(s)}\,ds. \end{aligned}$$

Different lower limits add a homogeneous solution. This construction is the ODE precursor of a Green-function representation.

First-Order Systems and Fundamental Matrices

Every higher-order ODE can be converted to a first-order system. For

$$y''+p(x)y'+q(x)y=r(x),$$

set

$$Y= \begin{pmatrix} y\\y' \end{pmatrix}.$$

Then

$$Y' = \begin{pmatrix} 0&1\\ -q&-p \end{pmatrix} Y + \begin{pmatrix} 0\\r \end{pmatrix}.$$

More generally,

$$Y'=A(x)Y+b(x).$$

A fundamental matrix $\Phi(x)$ solves

$$\Phi'=A(x)\Phi$$

and is invertible. The homogeneous solution is

$$Y(x)=\Phi(x)c.$$

For constant $A$,

$$\begin{aligned} Y(x) &= e^{A(x-x_0)}Y(x_0)\\ &\quad+ \int_{x_0}^{x} e^{A(x-s)}b(s)\,ds. \end{aligned}$$

This form connects scalar ODEs to matrix exponentials and finite-dimensional time evolution.

Initial-Value versus Boundary-Value Problems

An initial-value problem specifies enough data at one point to determine a local trajectory. For a regular second-order equation, that usually means $y(x_0)$ and $y'(x_0)$.

A boundary-value problem imposes conditions at more than one point, for example

$$y(0)=0, \qquad y(L)=0.$$

Boundary-value problems behave differently:

• a solution may not exist;
• several solutions may exist;
• a parameter may have to take special values;
• local initial-value uniqueness does not guarantee boundary solvability.

The page Boundary Conditions develops Dirichlet, Neumann, Robin, periodic, and matching conditions.

Eigenvalue Problems

In an ODE eigenvalue problem, a parameter $\lambda$ appears in the equation and nonzero solutions are allowed only when the boundary conditions are satisfied. The model problem

$$-y''=\lambda y, \qquad y(0)=y(L)=0$$

has nontrivial solutions only for

$$\lambda_n = \left( \frac{n\pi}{L} \right)^2, \qquad n=1,2,3,\ldots.$$

The corresponding functions are

$$y_n(x)=\sin\left(\frac{n\pi x}{L}\right)$$

up to normalization. Discreteness comes from the differential equation and both boundary conditions together, not from normalization.

The general operator viewpoint is Eigenvalue Problems.

Sturm–Liouville Preview

Many second-order eigenvalue equations can be written

$$-\frac{d}{dx} \left[ p(x)\frac{dy}{dx} \right] +q(x)y =\lambda w(x)y.$$

With suitable endpoint conditions and positive weight $w$, this is a Sturm–Liouville problem. Self-adjoint structure then explains real eigenvalues and weighted orthogonality:

$$\int_a^b y_m(x)^*y_n(x)w(x)\,dx =0.$$

This holds whenever $\lambda_m\ne\lambda_n$.

Regular and singular endpoint classifications, completeness, and weighted spaces belong to Sturm–Liouville Theory.

Schrödinger Equations as ODEs

For one particle in one dimension, the stationary Schrödinger equation is

$$-\frac{\hbar^2}{2m}\psi''(x) +V(x)\psi(x) =E\psi(x).$$

Equivalently,

$$\psi''(x) + \frac{2m}{\hbar^2} \bigl[E-V(x)\bigr]\psi(x) =0.$$

For each fixed $E$ and regular $V$, the local solution space is two-dimensional. Physical bound-state energies are selected only after domain, endpoint, matching, and square-integrability conditions are imposed.

Where $E-V$ is a positive constant, local solutions oscillate. Where it is a negative constant, local solutions are exponential. For varying potentials this observation motivates semiclassical approximations but is not, by itself, a global solution.

The canonical physics treatment is the Time-Independent Schrödinger Equation.

Ordinary and Singular Points

For

$$y''+P(x)y'+Q(x)y=0,$$

a point $x_0$ is ordinary when $P$ and $Q$ are analytic there. It is a regular singular point when

$$(x-x_0)P(x) \quad\text{and}\quad (x-x_0)^2Q(x)$$

extend analytically to $x_0$. More severe behavior gives an irregular singular point.

Near a regular singular point, the Frobenius ansatz is

$$\begin{aligned} y(x) &=(x-x_0)^s \sum_{n=0}^{\infty} a_n(x-x_0)^n, \\ &\qquad a_0\ne0. \end{aligned}$$

The lowest power gives the indicial equation for $s$. Repeated roots or roots differing by an integer can introduce logarithms. Radial Schrödinger, Bessel, and angular equations make these distinctions operational; see Bessel Functions.

Nondimensionalization

Choose a characteristic length $L$ and set

$$x=L\xi.$$

Then

$$\frac{d}{dx} = \frac{1}{L}\frac{d}{d\xi}, \qquad \frac{d^2}{dx^2} = \frac{1}{L^2}\frac{d^2}{d\xi^2}.$$

Rewriting an ODE in dimensionless variables:

• reduces the number of independent parameters;
• identifies perturbative regimes;
• improves numerical scaling;
• makes boundary conditions easier to compare.

Every term in the original dimensional equation must have the same units. A failed units check often reveals a missing scale or derivative factor.

Numerical Practice

Closed-form solutions are exceptional. Common numerical approaches include:

• adaptive Runge–Kutta methods for nonstiff initial-value problems;
• implicit methods for stiff equations;
• shooting methods for boundary-value and eigenvalue problems;
• finite-difference, finite-element, or spectral discretizations;
• matching inward and outward solutions near a stable interface.

For quantum eigenvalue ODEs:

• nondimensionalize before integrating;
• avoid integrating an exponentially growing forbidden-region solution over a long interval without stabilization;
• monitor both boundary mismatch and differential-equation residual;
• vary step size, domain size, and matching point independently;
• count nodes as a spectral diagnostic when the theorem applies;
• distinguish numerical normalization from satisfaction of boundary data.

Transfer matrices can become ill-conditioned when growing and decaying solutions coexist. Log-derivative or Riccati formulations can help, but they develop poles at zeros of the original solution. See ODE Solvers for algorithms.

Common Mistakes

• Applying superposition to a nonlinear equation.
• Quoting existence without checking uniqueness hypotheses.
• Treating a local solution as automatically global.
• Dividing by a leading coefficient at one of its zeros.
• Solving the differential equation while omitting initial or boundary data.
• Using normalization as a substitute for a boundary condition.
• Assuming every boundary-value problem has exactly one solution.
• Forgetting the repeated-root solution $xe^{rx}$.
• Using a Wronskian test on arbitrary functions as though they solved one common linear ODE.
• Ignoring singular points and endpoint classifications.
• Imposing finite-potential matching rules at a distributional singularity.
• Trusting a numerical boundary match without checking the ODE residual.

Exercises

1. Solve

   $$y'+2xy=2x, \qquad y(0)=0.$$

Solution

The integrating factor is

$$\mu(x)=e^{x^2}.$$

Multiplying the equation gives

$$\left( e^{x^2}y \right)' = 2xe^{x^2}.$$

Integrating,

$$e^{x^2}y=e^{x^2}+C,$$

so

$$y=1+Ce^{-x^2}.$$

The initial condition gives $0=1+C$, hence

$$y(x)=1-e^{-x^2}.$$

2. Let $y_1,y_2$ solve

   $$y''+p(x)y'+q(x)y=0.$$

   Derive Abel’s identity for their Wronskian.

Solution

Differentiate

$$W=y_1y_2'-y_1'y_2:$$ $$\begin{aligned} W' &= y_1y_2''-y_1''y_2\\ &= y_1(-py_2'-qy_2) -(-py_1'-qy_1)y_2\\ &= -p(y_1y_2'-y_1'y_2)\\ &=-pW. \end{aligned}$$

Solving this first-order equation gives

$$W(x) = W(x_0) \exp\left( -\int_{x_0}^{x}p(s)\,ds \right).$$

3. Find all $\lambda\in\mathbb R$ for which

   $$-y''=\lambda y, \qquad y(0)=y(L)=0$$

   has a nonzero solution.

Solution

If $\lambda=k^2>0$, then

$$y=A\sin(kx)+B\cos(kx).$$

The condition at zero gives $B=0$, and the condition at $L$ requires $\sin(kL)=0$. Thus

$$k=\frac{n\pi}{L}, \qquad n=1,2,3,\ldots.$$

If $\lambda=0$, then $y=Ax+B$, and both boundary conditions force $A=B=0$.

If $\lambda=-\kappa^2<0$, then

$$y=A\sinh(\kappa x)+B\cosh(\kappa x).$$

Again $B=0$, and $\sinh(\kappa L)\ne0$, so $A=0$. Therefore the only eigenvalues are

$$\lambda_n = \left( \frac{n\pi}{L} \right)^2.$$

4. Explain why the initial-value problem

   $$y'=2\sqrt{\lvert y\rvert}, \qquad y(0)=0$$

   is not unique by constructing infinitely many solutions on $x\ge0$.

Solution

For every $a\ge0$, define

$$y_a(x) = \begin{cases} 0,&0\le x\le a,\\ (x-a)^2,&x\ge a. \end{cases}$$

For $x<a$, both sides of the ODE vanish. For $x>a$,

$$\begin{aligned} y_a'(x) &=2(x-a) \\ &=2\sqrt{(x-a)^2} \\ &=2\sqrt{\lvert y_a(x)\rvert}. \end{aligned}$$

At $x=a$, both one-sided derivatives are zero, so $y_a$ is continuously differentiable and satisfies the equation there as well. Every $y_a$ obeys $y_a(0)=0$, and different waiting times $a$ give different solutions.

The function $2\sqrt{\lvert y\rvert}$ is continuous but not locally Lipschitz at $y=0$, so Picard–Lindelöf uniqueness does not apply.

References

• E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw–Hill, 1955.
• G. Teschl, Ordinary Differential Equations and Dynamical Systems, American Mathematical Society, 2012.
• W. E. Boyce, R. C. DiPrima, and D. B. Meade, Elementary Differential Equations and Boundary Value Problems, 11th ed., Wiley, 2017.
• C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers I, Springer, 1999.
• G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists, 7th ed., Academic Press, 2013.