Abstract: This paper presents a new method of weighted residuals called theMWR of mathematical programming,which is to seek the approximate solutionsof differential equations by the use of the mathematical programming and thebilateral inequality, as is different from other methods.By this method,it ispossible to find the upper bound and lower bound of the approximate solution.Moreover,this method possesses the advantage that the degree of accuracy ofthe approximate solutions may be controlled during the process of ca...