59-62] DIFFERENTIAL EQUATION OF PATH. 63 



61. Integration of the equation when f is a function 



r. Multiply both sides of the equat 

 the indefinite integral of u~~f, we have 



of r. Multiply both sides of the equation by -, and let &amp;lt;f&amp;gt;(u) be 



where A is constant. 



Suppose the initial condition is that the starting point is at a 

 distance c from the origin and the initial velocity is F in a direc 

 tion making an angle a with the radius vector. We have 

 h = Vc sin a. Also, by a well-known formula, if p is the perpen 

 dicular from the origin on the tangent at the point (u, 0), 



To determine the constant A, express that equation (1) holds 

 in the initial condition, we have 



1 



2c 2 sin 2 a F 2 c 2 sin 2 a r \c . 

 A is now determined, and equation (1) can be written 



where -x/r is a well-defined functional expression. 

 Integrating this equation we have 



e g== f du 



J Vi/r( 



where a is an arbitrary constant depending on the choice of the 

 initial line. 



62. Examples of Integration. 



1. Let the acceleration / be inversely proportional to the square of the 

 distance. 



We have f=p.u z so that the equation is 



Write I for A 2 //j, and vforu j, then the equation is 



c 



cPv 



