![]() ![]() If the A i are real then real-valued solutions are preferable. These functions make up a basis of the ODE's solutions. If z is a (possibly not real) zero of F( z) of multiplicity m and then is a solution of the ODE. This corresponds to the real-valued solution basis This has zeroes, i, − i, and 1 (multiplicity 2). This equation F( z) = 0, is the "characteristic" equation considered later by Monge and Cauchy. Plugging those values into gives a basis for the solution any linear combination of these basis functions will satisfy the differential equation. Solving the polynomial gives n values of z. Of the original differential equation are replaced by z k. So dividing by e z x gives the nth-order polynomial The first method of solving linear ordinary differential equations with constant coefficients is due to Euler, who realized that solutions have the form e z x, for possibly-complex values of z. Homogeneous linear ODEs with constant coefficients Information below provides methods for the solution of these differing ODEs:
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