59-62] DIFFERENTIAL EQUATION OF PATH. 63 



61. Integration of the equation when f is a function 



. Multiply both sides of the equa 

 the indefinite integral of ir z f, we have 



of r. Multiply both sides of the equation by -, and let <j>(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 V 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, 6), 



1 du\* 



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

 in the initial condition, we have 



1 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 -^ is a well-defined functional expression. 

 Integrating this equation we have 



du 







C du 

 a = I 



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=pu 2 so that the equation is 



Write I for h 2 /p. and v for u -i, then the equation is 



