I would like someone to introduce me to ODE and answer questions as they arise.

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ODE is the part of calculus where both integration as well as differentiation are used..
simple equation:
dy/dx = f(x) : degree 1, order 1.
ORDER of eqn. : order of highest derivative
DEGREE of eqn.: index or power or raised to the highest order derivative
e.g. (dy/dx)^3 + (d^2y/dx^2)^2 : order 2, degree 2 (not 3)
Now,
e.g. eqn: dy/dx = sin(x)
solution=> dy = sin(x). dx
=> y = integration[sin(x)]dx = -cos(x) + const.

One step ahead:
dy/dx = f(x,y) : order 1, degree 1
e.g dy/dx = sin(x)/{sec(y).tan(y)}
=> sec(y).tan(y) dy = sin(x) dx
=> int[sec(y).tan(y)] dy = int[sin(x)] dx
=> sec(y) = ...

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