Ordinary Differential Equations

An ordinary differential equation constrains an unknown function of one independent variable and its derivatives. A formula is not a complete solution until its interval, regularity, initial data, and any excluded or singular solutions are accounted for.

A rate law turns a mechanism into a prediction #

An ODE says how the present state controls its instantaneous rate of change. The initial condition identifies the state now; solving the equation converts that local rate rule into a trajectory through time.

The cooling model assumes one uniform object temperature, constant room temperature, and a constant lumped cooling-rate coefficient. The same pattern—state, rate law, initial state, trajectory—appears in motion, circuits, mixing, population models, and feedback control, but every application brings its own assumptions.

Equations, solutions, and initial data #

A first-order equation in normal form is

$$y'=f(t,y).$$

An initial-value problem (IVP) adds $y(t_0)=y_0$.

A local existence-and-uniqueness theorem is a contract, not decoration. A common version says that if $f$ is continuous near $(t_0,y_0)$ and locally Lipschitz in $y$ there—for example, if $f_y$ is continuous—then a unique local solution passes through the initial point.

If continuity or the Lipschitz condition fails, three different outcomes are possible:

• no solution;
• one solution;
• multiple solutions.

The theorem gives sufficient conditions. Failure of its hypotheses does not, by itself, decide which outcome occurs.

For a normalized linear second-order IVP

$$y''+p(t)y'+q(t)y=g(t),\qquad y(t_0)=a,\quad y'(t_0)=b,$$

continuity of $p,q,g$ on an interval containing $t_0$ gives a unique solution on that interval.

Separable equations and lost solutions #

If

$$y'=g(t)h(y),$$

then, where $h(y)\ne0$,

$$\frac{dy}{h(y)}=g(t)\,dt.$$

The condition $h(y)\ne0$ is load-bearing. Every root $h(c)=0$ must be checked separately, because $y(t)\equiv c$ may be an equilibrium solution discarded by division.

A robust workflow is:

1. list equilibrium solutions $h(y)=0$;
2. separate only on regions where $h(y)\ne0$;
3. integrate and keep absolute values where required;
4. impose initial data;
5. state the maximal interval before the solution reaches a singularity.

For the logistic equation $y'=y(1-y)$, the sign of the right-hand side sends positive trajectories toward $y=1$, while $y=0$ and $y=1$ are themselves constant solutions. They must be recorded before division by $y(1-y)$.

Nonuniqueness at a singular slope #

Consider the equation

$$y'=-\frac{\sqrt{1-y^2}}{\sqrt{1-x^2}}, \qquad y(0)=1.$$

The constant function $y\equiv1$ satisfies the IVP. Separating variables divides by $\sqrt{1-y^2}$, so it cannot recover that solution. Moreover the right-hand side is not locally Lipschitz in $y$ at $y=1$, and branch/domain restrictions matter. A nonconstant branch cannot be reported as the unique IVP solution.

Exact equations #

A differential form

$$M(x,y)\,dx+N(x,y)\,dy=0$$

is exact on a suitable domain if there is a potential $\Phi$ with

$$\Phi_x=M,\qquad \Phi_y=N.$$

Then the implicit solution is

$$\Phi(x,y)=C.$$

On a simply connected domain with continuous first partial derivatives, $M_y=N_x$ is sufficient for exactness. On a domain with holes, equal cross-partials need not yield a global single-valued potential.

To construct $\Phi$, integrate $M$ with respect to $x$:

$$\Phi(x,y)=\int M(x,y)\,dx+g(y),$$

then differentiate with respect to $y$ and solve for $g'(y)$ using $N$.

First-order linear equations #

For

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

choose the integrating factor

$$\mu(x)=e^{\int p(x)\,dx}.$$

Because $\mu'=\mu p$,

$$(\mu y)'=\mu y'+\mu p\,y=\mu q.$$

Hence

$$y(x)=\frac{1}{\mu(x)} \left(C+\int \mu(x)q(x)\,dx\right).$$

The method works on intervals where $p$ and $q$ are continuous. An integrating factor may have different expressions on disconnected intervals.

Worked example #

Solve

$$y'+2xy=e^{-x^2}x\sin x.$$

The integrating factor is

$$\mu=e^{\int2x\,dx}=e^{x^2}.$$

Therefore

$$(e^{x^2}y)'=x\sin x.$$

Integration by parts gives

$$\int x\sin x\,dx=\sin x-x\cos x,$$

so

$$\boxed{y=e^{-x^2}\left(C+\sin x-x\cos x\right)}.$$

Substitution differentiates the product and recovers the original equation, which is the decisive check on the integrating-factor calculation.

Bernoulli equations #

A Bernoulli equation is

$$y'+p(x)y=q(x)y^n,\qquad n\ne0,1.$$

On an interval where $y\ne0$, set $v=y^{1-n}$. Since

$$v'=(1-n)y^{-n}y',$$

multiplying the ODE by $(1-n)y^{-n}$ gives a linear equation:

$$v'+(1-n)p(x)v=(1-n)q(x).$$

Check $y=0$ separately when the original equation makes sense there. Describing $y=0$ as the case “$C=\infty$” is not legitimate: infinity is not a value of the integration constant in the derived solution family.

Higher-order linear equations #

A linear equation has the form

$$a_n(t)y^{(n)}+\cdots+a_1(t)y'+a_0(t)y=g(t), \qquad a_n(t)\ne0.$$

After division by $a_n$, existence and uniqueness are controlled on intervals where all normalized coefficients and $g$ are continuous.

For the homogeneous constant-coefficient equation

$$a_ny^{(n)}+\cdots+a_1y'+a_0y=0,$$

try $y=e^{rt}$. This produces the characteristic polynomial

$$a_nr^n+\cdots+a_1r+a_0=0.$$

The solution contributions are:

• real root $r$ of multiplicity $m$:

  $$e^{rt},\,te^{rt},\,\ldots,t^{m-1}e^{rt};$$

• conjugate roots $\alpha\pm i\beta$, each of multiplicity $m$:

  $$t^k e^{\alpha t}\cos\beta t,\quad t^k e^{\alpha t}\sin\beta t,\quad 0\le k<m.$$

The number of linearly independent basis functions must equal the order of the equation.

Nonhomogeneous equations and superposition #

For a linear operator $L$,

$$L[y]=g,$$

the general solution is

$$y=y_h+y_p,$$

where $L[y_h]=0$ and $L[y_p]=g$.

This follows because

$$L[y_h+y_p]=L[y_h]+L[y_p]=g.$$

The forcing $g$ is not itself a solution; confusing the input with the response is a category error.

Variation of parameters finds $y_p$ for general forcing. Undetermined coefficients is faster for polynomial, exponential, sine, and cosine inputs, but only after accounting for resonance.

If the naive trial duplicates a homogeneous mode, multiply the trial by enough powers of $t$ to make it linearly independent. Resonance is therefore a linear-dependence problem, not a mysterious extra rule.

For an ideal undamped oscillator driven at its natural frequency, the resonant particular solution grows proportionally to $t$. Real damping changes the response and must not be omitted when making physical claims.

Linear systems #

A constant system is

$$x'=Ax.$$

If $Av=\lambda v$, then

$$x(t)=e^{\lambda t}v$$

is a solution. Distinct eigenvectors generate independent modes.

A repeated eigenvalue does not automatically provide enough eigenvectors. If $A$ is defective, find a generalized eigenvector $w$ satisfying

$$(A-\lambda I)w=v.$$

A Jordan-chain pair of solutions is then

$$e^{\lambda t}v, \qquad e^{\lambda t}(tv+w).$$

Writing four arbitrary constants for a two-dimensional repeated-root system is impossible: a first-order $2\times2$ system has a two-dimensional solution space.

For nonhomogeneous systems

$$x'=A(t)x+g(t),$$

a fundamental matrix $\Phi(t)$ gives variation of constants:

$$x(t)=\Phi(t) \left(c+\int \Phi(t)^{-1}g(t)\,dt\right).$$

Why eigenvectors matter: two coupled rooms #

Consider two thermal zones with equal effective heat capacities that exchange heat but, in this simplified model, lose no heat outdoors:

$$T_1'=-k(T_1-T_2), \qquad T_2'=k(T_1-T_2).$$

In matrix form,

$$\begin{bmatrix}T_1'\\T_2'\end{bmatrix} = \begin{bmatrix}-k&k\\k&-k\end{bmatrix} \begin{bmatrix}T_1\\T_2\end{bmatrix}.$$

The eigenvector $(1,1)$ has eigenvalue $0$: equal temperatures do not change, and the average is conserved. The eigenvector $(1,-1)$ has eigenvalue $-2k$: a temperature difference decays exponentially.

If $k=0.1\ \mathrm{min^{-1}}$, $T_1(0)=30^\circ\mathrm C$, and $T_2(0)=20^\circ\mathrm C$, then

$$T_1(t)=25+5e^{-0.2t}, \qquad T_2(t)=25-5e^{-0.2t}.$$

After ten minutes they are approximately $25.68^\circ\mathrm C$ and $24.32^\circ\mathrm C$. Eigenvectors reveal two independent physical patterns—a conserved average and a decaying difference—that are hidden when the coupled equations are read one row at a time.

Models and singular initial conditions #

A differential equation inherits the validity limits of the physical law used to derive it. For example, a force law containing $1/v$ is undefined at $v=0$. Algebraically finding a curve that approaches $v=0$ does not make an IVP posed at $v(0)=0$ valid. The model must be reformulated or its domain restricted before applying an existence theorem.

Likewise, exponential decay in an $RL$ circuit assumes fixed $R,L$, a specified voltage input, and an ideal lumped model. State these assumptions before interpreting long-time behavior.

Verification checklist #

For every proposed ODE solution:

1. substitute it into the original equation, not only a transformed equation;
2. check all initial or boundary data;
3. identify divisions and substitutions that may have removed solutions;
4. state the interval and singular points;
5. confirm that the number of arbitrary constants matches the order;
6. for systems, verify eigenvectors or generalized eigenvectors directly;
7. separate mathematical behavior from model assumptions.

Failure modes #

• Dividing by $y$, $1-y^2$, or another solution-dependent factor without checking its zeros.
• Writing a singular solution as a limiting “infinite constant.”
• Claiming uniqueness when the vector field is not locally Lipschitz at the initial point.
• Treating vertical curves as solutions of an equation posed for $y(x)$.
• Omitting absolute values introduced by logarithmic integration.
• Using a characteristic-root formula across a point where the leading coefficient vanishes.
• Forgetting the $t$ factor for repeated roots or resonance.
• Treating a repeated eigenvalue as if it automatically supplied a full eigenbasis.
• Reporting only a particular solution to a nonhomogeneous equation.
• Applying a physical ODE at an initial state where its own right-hand side is undefined.