Ordinary Differential Equations by L. S. Pontryagin and A. J. Lohwater (Auth.)

By L. S. Pontryagin and A. J. Lohwater (Auth.)

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Rows of the matrix are to be multiplied in order that their sum be zero. By writing the sum of the elements of the jth column, we obtain the equality hozf-l\to) + &l4 n - 2 ) (

The function

If the characteristic polynomial L(p) of the equation L(p)z = 0 (6) [see (1) and (4)] has no multiple roots, if its roots are λχ, λ2, · · . , λ η , and if we set zi = β λιί , 22 = e^\ . . , zn = eV, (7) then for any complex constants c1, c2, . . , c n , the function z = clzl + c2z2 H h cnzn (8) is the solution of equation (6). This solution is the general solution in the sense that every solution of equation (6) can be obtained from (8) by proper choice of the constants c1, c2, . . , cn. Here the constants c1, c 2 ,.

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