6 1. Differential Equations simply impossible to write any explicit solutions in terms of functions that we already know and have a notation for. This does not mean that there are no solutions to these equations, as we can sometimes prove analytically that solutions exist and can describe their behavior either qualitatively or approximate it graph- ically even when those solutions cannot be written in terms of the functions that we have already named. The equations do not have to be very complicated in order to be in this category. The simplest sort of differential equation, which you surely solved many times in your calculus class, is an equation of the form dy dx = f(x). (This is an ordinary, linear differential equation for the unknown func- tion y, but it is nonautonomous because f(x) is one of the coefficients in the equation and it depends on x.) The general solution to this equation is y = F (x) + C where F (x) is any anti-derivative of f(x) and C is any constant. For instance, if f(x) = 2x, then the general solution is y = x2 + C. Yet, if the function on the right-hand side of the equation is f(x) = sin(x2) then you cannot write a formula for the solution unless you make use of integration. Indeed, the Fundamental Theorem of Calculus tells us that F (x) = x 0 sin(t2) dt is one such antiderivative. However, to compute the value of this function F (x) will require you to numerically approximate the value of the integral using a Riemann Sum or Simpson’s Rule. Even if you knew the values of the sine function exactly, you would only be able to approximate values for F . In most advanced courses on differential equations, it is impor- tant to be able to analyze equations whose solutions can be shown to exist even though they cannot be written exactly. Numerical approx- imation methods, similar to using a Riemann Sum or Simpson’s Rule to approximate the value of an integral, are often used to compute values for the functions in order to graph or animate them. That will not be the case here, however. As we will see, in soliton theory it is possible to produce many exact formulas for solutions using algebra and geometry.
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