Solution: y’ = e^(x-y) ⟹ dy/dx = e^x/e^y |*dx*e^y ⟹ e^y*dy = e^x*dx |∫() ⟹ ∫e^y*dy = ∫e^x*dx ⟹ y = ln|e^x+C| ⟹ y’ = e^x/(e^x+C) Does the solution fit into the differential equation? left side: e^x/(e^x+C) right side: e^(x-ln|e^x+C|) = e^x/e^(ln|e^x+C|) = e^x/(e^x+C) left side = right side everything is o.k.
One of those "separable"-ones, lol, probably...that N, M-thing or whatever...rewrite it as dy/dx = e^x (e^{-y}), etc, and then put all the x's on one side, all the y's on the other, etc...I forgot how you do all that mindless calculation, lol...
The first method is a bit excessive. You get to a separable equation, but the original equation was already separable.
Thanks for the video as always.
Shouldn't you address that k > - e^x is necessary for the solution to make sense?
Solution:
y’ = e^(x-y) ⟹
dy/dx = e^x/e^y |*dx*e^y ⟹
e^y*dy = e^x*dx |∫() ⟹
∫e^y*dy = ∫e^x*dx ⟹
y = ln|e^x+C| ⟹ y’ = e^x/(e^x+C)
Does the solution fit into the differential equation?
left side: e^x/(e^x+C)
right side: e^(x-ln|e^x+C|) = e^x/e^(ln|e^x+C|) = e^x/(e^x+C)
left side = right side everything is o.k.
One of those "separable"-ones, lol, probably...that N, M-thing or whatever...rewrite it as dy/dx = e^x (e^{-y}), etc, and then put all the x's on one side, all the y's on the other, etc...I forgot how you do all that mindless calculation, lol...
y' = e^x/e^y
∫e^y dy = ∫e^x dx
e^y = e^x + c
y = ln(e^x + c)
And k >0.