The approximate integration of differential equations

There are two interesting and useful methods for obtaining the approximate solution of differential equations  

I Integration in series. When a function can be developed in a series of converging terms, arranged in powers of the independent variable, an approximate value for the dependent variable can easily be obtained. The degree of approximation attained obviously depends on the number of terms of the series included in the calculation. The older mathematicians considered this an underhand way of getting at the solution, but, for practical work, it is invaluable. As a matter of fact, solutions of the more advanced problems in physical mathematics are nearly always represented in the form of an abbreviated infinite series. Finite solutions are the exception rather than the rule. 

Examples.—(1) It is required to find the solution dyjdx = y, in series. Assume that y has the form 

If x is not zero, this equation is satisfied when the coefficients of x become zero. This requires that 

Put a for the arbitrary constant so that the final result is y = aex. That this is a complete solution, is proved by substitution in the original equation. We 

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