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) = ...

Solution Summary

This provides a brief introduction to ordinary differential equations

Solutions to First-Order Ordinary Differential Equations. ... We solve several first-order ordinary differential equations. Implicit equations are examined. ...

Introduction to Ordinary Differential Equations. ... An ordinary differential equation (ODE) is an equation that involves derivatives, but no partial derivatives. ...

Solving two independent ordinary differential equations. ... The solution of this particular Ordinary Differential Equation (ODE) is attained by making the guess. ...

... The difference between ordinary differential equations, which we often refer to as ODEs, and partial differential equations, which we often refer to as PDEs ...

Systems of Ordinary Differential Equations (4 Problems). Problem 8.1 (Prob. 11, p.251) Solve the following system of equations. { x' = y { y' = -x. ...