Summary of Solution Methods for ODE

We started this chapter by delineating the two fundamental types of equations, either nonlinear or linear. We then introduced the few techniques suitable for nonlinear equations, noting the possibility of so-called singular solutions when they arose. We also pointed out that nonlinear equations describing model systems usually lead to the appearance of implicit arbitrary constants of integration, which means they appear within the mathematical arguments, rather than as simple multipliers as in linear equations. The effect of this implicit constant often shows up in startup of dynamic systems. Thus, if the final steady state depends on the way a system is started up, one must be suspicious that the system sustains nonlinear dynamics. No such problem arises in linear models, as we showed in several extensive examples. We emphasized that no general technique exists for nonlinear systems of equations. 

The last and major parts of this chapter dealt with linear equations, mainly because such equations are always solvable by general methods. We noted that forced equations contain two sets of solutions the particular solutions [related directly to the type of forcing function /(x)] and the complementary solution [the solution obtainable when fix) = 0], so that in all cases y(x) = ycix) -I-ypix). We emphasize again that the arbitrary constants are found (in conjunc- 

Linear homogeneous equations containing nonconstant coefficients were not treated, except for the elementary Euler-Equidimensional equation, which was reduced to a constant coefficient situation by letting x = exp(r). In the next chapter, we deal extensively with the nonconstant coefficient case, starting with the premise that all continuous solutions are in fact representable by an infinite series of terms, for example ejqj (jc) = 1 + x + x /2 + x /3 + . This leads to a formalized procedure, called the Method of Frobenius, to find all the linearly independent solutions of homogeneous equations, even if coefficients are nonconstant. 

Amundson, N. R. Mathematical Methods in Chemical Engineering, Matrices and Their Application. Prentice Hall, Englewood Clifik, New Jersey (1966). 

Hildebrand, F. B. Advanced Calculus for Applications. Prentice Hall, Englewood Cliffs, New Jersey, pp. 5, 34 (1962). 

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