Mechanics

0. Mathematical Preliminaries

2. Differential Equations
2-1. Separation of variables

__There are many methods developed for solving differential equations. Here, we only describe convenient method by Separating variables. First, check the first degree differential equations.

Illustrated Example 2-1. Solve the following differential equations.

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Explanations and Answers: 2-1. (1) Multiplying dx, y² on both sides, we have y²dy = xdx. Variables are separated. Integrating,

2-1. (2) Separating variables and integrating, we have the result: y^-1 = x²/2 + C. Applying the initial condition, 1 = 1/2 + C
C = 1/2. Therefore, 1/y + x²/2 + 1/2= 0 or,

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__The first order differential equation of type:

,where a0.

If F(t) = 0, this type appears as exponential decay in many physical phenomena, such as the cooling of a hot object's temperature,
the some types of chemical reactions, the decay of radioactive substance, etc. If F(t) = constant, the solution is obtained to be
x(t) = Ae^-t/a+ B, where A, B are constant. In general case, F(t)const., and even a, b may be a function, the method of variation of constants described in the next section can be utilized, or numerical method may be adopted to plot the variations in x(t).