So my colleague K stops by my office today and asks if I’m busy.

“No. What’s up?”

“I’ve got a thermodynamics problem and I’m looking for a way to solve it,” says K, who is a chemical engineer by trade. He proceeds to set the stage, explaining that this is a problem about pressure and temperature changes in a fire extinguisher when in use. Starting with the ideal gas law, the conservation of energy, an assumption on the nozzle, and the multivariate chain rule, K lays out a sequence of equations. Six blackboards later, he ends with a pair of differential equations:

“The problem,” concludes K, “is to solve these equations for *P* and *T*, where the other variables — the *V*s and *R*s and *C*s — are constants.”

“Well… we can use the first equation to express *P* in terms of *T*, and see what the solutions look like in phase space,” I suggest.

So we separate ignore the independent variable and separate the remaining variables and go to town:

which yields *P* in terms of *T* as

.

“Okay…” I say. “Well, since the second equation only involves *P* but not its derivative, we could substitute the formula for *P* in it, and reduce the second equation to a single differential equation involving *T*.”

“Oh!” I say, more excited. “This is now a first-order autonomous ODE, so it’s separable too!”

.

Incredibly, the integral on the left-hand-side can be evaluated with a simple substitution (a fact at which I, admittedly, giggled with glee), so we get

.

“So we’ve got (big) *T* solved implicitly in terms of (little) *t*,” I say, “so let’s unshackle it explicitly.”

“And you have a *T*!” I exclaim, surprised that any of that worked. “Now we can substitute into our implicit formula for *P* to get it to.”

“By way of checking,” I add, anxious to make sure I didn’t make some bone-headed algebra mistake along the way, “we can check the initial conditions on T…”

“and P…”

“and the first differential equation…” (after furiously scribbling for a while) “…checks out, since both sides simplify to

and each side of the second differential equation…” (more furious scribbling) “… simplifies to”

…so, yeah, we have a solution!” I conclude triumphantly.

K stares at the work, inspecting the derivation and double-checking some thermodynamical facts in his head. Finally, he pauses, scribbles some notes down in his binder, and turns to me.

“Thanks,” K says, adding, “I *knew* it was something simple.”

It was something simple.