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In elementary ODE textbooks, an early chapter is usually dedicated to first order equations. It is very common to see individual sections dedicated to separable equations, exact equations, and general first order linear equations (solved via an integrating factor), not necessarily in that order.

Common practical applications in these texts include population growth/decay, mixing problems, draining tank/Torricelli's Law problems, projectile motion, Newton's Law of Cooling, orthogonal trajectories, melting snowball type problems, certain basic circuits, growth of an annuity, and logistic population models. (This is just off the top of my head so maybe I am missing other popular ones.) However, all of these end up as separable or first order linear problems and are solved accordingly.

Are there practical applications that lead to first order ODEs which are (exclusively) exact equations?

Edit: To clarify, I am not saying that exact equations are never useful. I am simply inquiring about their relevance/applicability in the very particular context mentioned above.

To put the question another way, can you briefly state (e.g., in the form of an exercise that would appear in popular undergrad ODE books like Boyce & DePrima; Zill; Nagle/Saff/Snider; Edwards & Penney; etc.) an application problem modeled by a first order exact ODE (which is not separable or linear) and that is solvable by hand? I've looked in the dozen or so ODE textbooks on my shelf and none of them contain such a problem. I find that absence curious.

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Sorry, it is the TWO YEAR ANNIVERSARY of the question, not one year! –  JohnD Dec 19 '14 at 16:24

4 Answers 4

Wikipedia references:

<quote> Streamlines are a family of curves that are instantaneously tangent to the velocity vector of the flow. These show the direction a massless fluid element will travel in at any point in time. </quote>
Consider the velocity field $(u,v)$ of a two-dimensional incompressible flow. Let the family of curves be given by $\;\psi(x,y) = c$ . The velocity vectors are tangent to these as shown for one of them in the following picture.

enter image description here

Thus, along the curve $\psi(x,y) = c$ , the following equations hold: $$ \left. \begin{array}{l} \frac{dy}{dx} = \frac{v}{u} \\ d\psi = 0 = \frac{\partial \psi}{\partial x} dx + \frac{\partial \psi}{\partial y} dy \end{array} \right\} \qquad \Longrightarrow \qquad \frac{dy}{dx} = - \frac{\partial \psi / \partial x}{\partial \psi / \partial y} = \frac{v}{u} $$ Hence, apart from a constant: $$ u = \frac{\partial \psi}{\partial y} \qquad ; \qquad v = - \frac{\partial \psi}{\partial x} $$ But the flow is incompressible, so: $$ \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0 \qquad \Longrightarrow \qquad \frac{\partial^2 \psi}{\partial x\, \partial y} = \frac{\partial^2 \psi}{\partial y\, \partial x} $$ Herewith the conditions for an exact differential equation are fulfilled. Now solve $\psi$ from: $$ v\, dx - u\, dy = 0 $$ Example. As taken from : Find the velocity of a flow . $$ u = -\frac{y}{x^2+y^2} \qquad ; \qquad v = \frac{x}{x^2+y^2} $$ Then: $$ v\, dx - u\, dy = \frac{x\,dx + y\,dy}{x^2+y^2} = \frac{d\left( x^2+y^2 \right)}{x^2+y^2} = 0 \qquad \Longrightarrow \qquad x^2 + y^2 = c $$ It is concluded that the streamlines of this flow are circles.

Example. Somewhat related to the above one. $$ u = \lambda\,x \qquad ; \qquad v = \lambda\,y $$ Then, assuming that $\; x\ne 0$ (i.e. $\,x=0\,$ as a special case) : $$ v\, dx - u\, dy = 0 \quad \Longleftrightarrow \quad \frac{y\,dx - x\,dy}{x^2} = - d(y/x) = 0 \quad \Longrightarrow \quad y = c\, x $$ An integrating factor has been used. It is concluded that the streamlines of this flow are straight lines through the origin.

Wikipedia reference:

The electric potential at a point $\vec{r}$ in a two-dimensional static electric field $\vec{E}$ is given by the line integral: $$ V = - \int_C \vec{E}\cdot d\vec{r} = - \int_C \left(E_x\, dx + E_y\, dy\right) $$ where $C$ is an arbitrary path connecting the point with zero potential to $\vec{r}$. It follows that: $$ E_x = - \frac{\partial V}{\partial x} \qquad ; \qquad E_y = - \frac{\partial V}{\partial y} $$ The integral is zero if the path is closed. Then Green's theorem tells us: $$ \oint \left( E_x\, dx + E_y\, dy \right) = \iint \left( \frac{\partial E_y}{\partial x} - \frac{\partial E_x}{\partial y} \right) dx\,dy = - \iint \left( \frac{\partial^2 V}{\partial x \, \partial y} - \frac{\partial^2 V}{\partial y \, \partial x}\right) dx\,dy = 0 $$ Thus establishing once more the conditions for solvability of the exact differential equation: $$ E_x \, dx + E_y \, dy = 0 $$ Solving this ODE results in the iso-lines $\,V(x,y) = c\,$ of the electric potential $\,V$ .

Example. An infinitely long and infinitely thin charged wire perpendicular to the plane and intersecting it in the origin. Apart from constants: $$ (E_x,E_y) = \frac{(x,y)}{r^2} \quad \Longrightarrow \quad E_x\, dx + E_y\, dy = \frac{x\,dx + y\,dy}{r^2} = 0 \quad \Longrightarrow \quad x^2+y^2 = c $$ The equipotential lines are circles.
Can of worms: special cases, and a singularity at the origin in all of the examples.

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+1 This is the kind of answer/application I am after. –  JohnD Dec 21 '14 at 20:49

I am not quite sure that this is exactly an example you are looking for, but still.

Consider the Lotka--Volterra system on the plane $$ \dot x=x(a_1x+b_1y+c_1)=P(x,y),\\ \dot y=y(a_2x+b_2y+c_2)=Q(x,y).\tag{1} $$

The following theorem is true: System $(1)$ does not have limit cycles. The proof is based on the Dulac's criterion that the expression $$ \frac{\partial}{\partial x}(BP)+\frac{\partial}{\partial y}(BQ)\tag{2} $$ has a definite sign. Here $B=x^{k-1}y^{h-1}$ is an integrating factor and $k,h$ depend on the parameters of the model.

It is possible that expression $(2)$ can be zero, in this case (and this is your example) the equation $$ \frac{dy}{dx}=\frac{Q(x,y)}{P(x,y)} $$ admits the integration factor $B$ and, after multiplication by $B$, is exact, hence admitting an analytic integral. And in this case therefore the phase plane consists of closed orbits.

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I believe that the question being asked is not about exact solutions to differential equations but a certain class of differential equations which are termed (exact differential equations). Two of the reason that they are useful is because there is a known method for solving them, and exact equations include a certain class of non-linear equations which are unsolvable by the methods you've learned for separable and linear equations. The more techniques you know and the more types of equations you know how to message information out of the more useful you will find differential equations for studying the real world (or for understanding pure mathematics). Hope this was helpful.

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I am looking for practical, physical phenomena which are modeled by exact ODEs (but not any of the other standard classes of ODEs). This is my particular sense of "useful". –  JohnD Feb 22 '14 at 22:34
@JohnD: Potential theory is a rich source of exact equations, e.g. Laplace equation. –  Narasimham Dec 26 '14 at 15:28


A cylindrical pylon two meters across stands in the middle of a slow, deep river. Let's use a cylindrical coordinate system centered on the pylon, with radial, angular, and vertical coordinates $r$, $\theta$, and $z$, measuring all distances in meters. The water in the river moves with velocity $$\left(1 - \frac{1}{r^2}\right) \cos(\theta) \frac{\partial}{\partial r} - \left(1 + \frac{1}{r^2}\right) r \sin(\theta) \frac{\partial}{\partial \theta}.$$

Write an equation describing the path of a diatom drifting down the river. There are many possible paths, but you should choose the one that approaches the line $$\begin{align} r \sin(\theta) & = 1/2 \\ z & = -2 \end{align}$$ when the diatom is very far from the column. Your equation only needs to describe where the diatom goes; it does not have to specify where the diatom is at each time.


  • Whenever I have to solve an exact ODE, what I'm really doing is finding the kernel foliation of an exact 1-form (or, equivalently, the integral curves of an exact vector field). So, if you want to find applications of exact ODEs, my advice is to look for applications of integral curves of exact vector fields. The problem above is a classic example.

  • As Han de Bruijn pointed out, there's plenty more problems where this came from. The general form is, "Find the streamlines of the two-dimensional flow with velocity field $\nabla \psi$."

  • Sorry about notating the coordinate basis vectors as derivative operators, the way a differential geometer would do it. I'm not sure what notation one uses for this when writing a textbook for engineers.

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+1 Nice answer and upvote. –  JohnD Dec 22 '14 at 6:06
It may be due to my old brain, but I'm at lost with this example. E.g. how can a differentiation be a velocity? –  Han de Bruijn Dec 22 '14 at 19:46
@HandeBruijn, one way to define a velocity on a manifold—that is, a tangent vector—is by saying how fast each function changes when you move with that velocity. This gives an identification between tangent vectors and derivative operators. You can find more details about this point of view in many introductory books on differential geometry; my favorite is the first chapter of Wald's General Relativity. –  Vectornaut Dec 26 '14 at 1:26
@JohnD, I'm glad you like it! If I find another good problem in this vein, I'll post that as well. –  Vectornaut Dec 26 '14 at 1:40

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