10.46 If we let x = 

 included in the form 



NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS 22Q 



x' = oct, y = x 3t y f = X*, equations 10.45 are 



2, - - ,*,*), 



This is the final standard form to which it will be supposed the differential 

 equations are reduced. 



10.50 Definition of a Solution of Differential Equations. For simplicity in 

 writing, suppose the differential equations are two in number and write them in 

 the form 



I. 





where / and g are known functions of their arguments. Suppose x = a, y = b 

 at t = o. Then 



*-*<, 



= *(0, 



is the solution of (i) satisfying these initial conditions if < and ^ are 



such functions that 



<o = a 



-/<*,**, 



the last two equations being satisfied for all o ^ t ^ T, where T is a positive con- 

 stant, the largest value of t for which the solution is determined. It is not neces- 

 sary that and ^ be given by any formulas -it is sufficient that they have 

 the properties defined by (3). Solutions always exist, though it will not.be 

 proved here, iff and g are continuous functions of t and have derivatives wth respect 

 to both x and y. 



10.61 Geometrical Interpretation of a Solution of Differential Equations.. 

 Geometrical interpretations of definite integrals have been of great value not 

 only in leading to an understanding of their real meaning but also in suggesting 



