Differential Equations Example confusion

July 17, 2019 9:04 AM

I'm having a trouble following one step of an example in a differential equations textbook. It's an example demonstrating solving a Cauchy-Euler equation by using a substitution to convert it to an equation with constant coefficients.

I know its probably something pretty straightforward, but I'm not seeing it. Unfortunately I'm reviewing on my own right now, so I don't have someone else to ask. Can someone look at the step highlighted in green and explain to me what's going on there?

I figure I'll probably smack myself in the forehead when someone does, but at least I won't be hung up anymore.

posted by Reverend John to Education (3 answers total)

Best answer: Recall that the chain rule states:

d/dx f(g(x)) = f'(g(x)) * g'(x)

Here, the function f is y, and the function g(x) = t = ln(x).

Then d/dx (y'(ln(x)) = y''(ln(x)) * 1/x = y''(t) * 1/x.
posted by YoloMortemPeccatoris at 9:25 AM on July 17, 2019 [1 favorite]

Best answer: It took me a moment to see it, but it's the Chain Rule: d/dx f= dt/dx df/dt. Here, we apply it to f=dy/dt and t=ln x, so
d/dx (dy/dt) = dt/dx d^2y/dt^2 = 1/x d^2y/dt^2.
posted by ectabo at 9:25 AM on July 17, 2019 [1 favorite]

Response by poster: Alright, yeah, I'm buying it. Don't know why I couldn't see it, but I think it would have helped if they had written the second step in the green box as two steps:

d/dx (dy/dt)
= d^2y/dt^2 dt/dx
= d^2y/dt^2 1/x

though it seems obvious now.

Thanks everyone!
posted by Reverend John at 11:36 AM on July 17, 2019