- #1
cmkluza
- 118
- 1
My current understanding of differential equations is extremely shaky, and my vocabulary is probably very incorrect, but I'm curious about something I've recently seen in some Khan Academy videos (specifically this one) and in other situations with differential equations. It seems that the following is true:
[tex]\frac{d}{dx} \psi (x, y) = \frac{∂\psi}{∂x}\frac{dx}{dx} + \frac{∂\psi}{∂y}\frac{dy}{dx} [/tex]
Why is this? The video on Khan Academy shows a "proof" by which he assumes ##\psi## can be represented by a sum of products of functions of x and y (##\psi = f_1(x)g_1(y) + ... + f_n(x)g_n(y)##). What is the basis of this "proof"? Why could some function of x and y, psi, be displayed as products of separate functions of x or y?
I feel as though following the flow of videos on Khan Academy isn't explaining differential equations well-enough for me. Can anyone here recommend where else I might look to learn about differential equations?
Thanks for any help!
[tex]\frac{d}{dx} \psi (x, y) = \frac{∂\psi}{∂x}\frac{dx}{dx} + \frac{∂\psi}{∂y}\frac{dy}{dx} [/tex]
Why is this? The video on Khan Academy shows a "proof" by which he assumes ##\psi## can be represented by a sum of products of functions of x and y (##\psi = f_1(x)g_1(y) + ... + f_n(x)g_n(y)##). What is the basis of this "proof"? Why could some function of x and y, psi, be displayed as products of separate functions of x or y?
I feel as though following the flow of videos on Khan Academy isn't explaining differential equations well-enough for me. Can anyone here recommend where else I might look to learn about differential equations?
Thanks for any help!