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 Vs and Rs and Cs — 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 $t$ 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 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.”