Title:

Author:

Mechanics
Copyright:

T. Tsujioka
Advanced Seminar    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.  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, __The first order differential equation of type: ,where a 0.
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).          