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*, *y*^{2} on both sides, we have *y*^{2}*dy* = *xdx*. Variables are separated.
Integrating,

2-1. (2) Separating variables and integrating, we have the result: *y*^{-1} = *x*^{2}/2 + *C*. Applying the initial condition, 1 = 1/2 + *C*

*C* = 1/2. Therefore, 1/*y* + *x*^{2}/2 + 1/2= 0 or,

__The first order differential equation of type:

,whereIfa0.

the some types of chemical reactions, the decay of radioactive substance, etc. If