Question

Find the particular solution (solved for y) for the differential equation dy/dx=2x/e^(2y) satisfying y(0)=1.

Answer 1

e^-2y dy = 2 x dx

e-2y /-2 = x^2 + c

e^-2y = -2 x^2 + c

-2y = ln (-2 x^2+c)

when x = 0, y = 1
-2 = ln(c)
e^-2 = c

-2 y = ln (-2x^2+e^-2)

y = -(1/2) ln(-2x^2+1/e^2)

Answer 2

Can you explain why it's simplified to e^-2ydy=2xdx. I don't understand why it's e^negative 2y