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The first and the third boundary value problems for the ordinary differential equation of the second order are solved by the net method provided that the coefficients as well as the right hand side may have a finite number of discontinuities. Some smoothness assumptions are made on the intervals of continuity. A finite difference analogue of the boundary value problem is constructed for the net which includes the points of discontinuity and is equidistant on each interval of continuity (but generally with different steps in various intervals). The existence and uniqueness of the solution of the discretised problem as well as the estimate max $E_i=O(h^{3/2})$ for the difference $E_i$ of the approximate and the exact solution are proved, $h$ being the maximal step of the net.
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