The Approximate Solution of Linear Differential Equations 



By MARION C. GRAY and S. A. SCHELKUNOFF 



Linear differential equations with variable coefficients occur in man>' fields of 

 applied mathematics: in the theories of acoustics, elastic waves, electromagnetic 

 waves in stratified media, nonuniform transmission lines, wave guides, antennas, 

 wave mechanics. The "Wave Perturbation" method described in greater detail 

 elsewhere^ is particularly useful in those ranges of the independent variable in 

 which the "WKB Approximation" is not sufficiently accurate. The present 

 paper endeavors to illustrate the remarkable accuracy of this method, particu- 

 larly when compared with Picard's method. 



I. Introduction 



IN A recent paper^ the approximate solution of linear differential equations 

 by a wave perturbation method was described. When the method was 

 appHed to equations whose exact solutions were known we were greatly 

 impressed by the rapidity of convergence of the successive approximations. 

 Hence the purpose of this note is to present some illustrations in the hope 

 that others may be interested and may find the proposed method an im- 

 provement on those now in use. 



In essence the wave perturbation method dates back to Liouville-, but 

 in his memoires he was interested in a problem of heat conduction .involving 

 a non-homogeneous differential equation with homogeneous boundary 

 conditions, whereas we consider a homogeneous equation 



/' = F(x)y (1) 



with non-homogeneous initial conditions 



y{a)=\,y{a) = (2a) 



or 



y{a) = y(a) = 1, (2b) 



the solution being desired in an interval a ^ x ^ b. Since the solution 

 for any assigned initial or boundary conditions can be expressed as a linear 

 combination of the solutions satisfying (2a) and (2b) we have not imposed 

 any real limitation. 



II. Theory 



Comparison of the wave perturbation method with Picard's method 

 (which is essentially a linear perturbation method) is particularly instruc- 



350 



