![]() Then the operator in brackets is a linear operator. Thus, if L denotes this operator and if f and g are functions and c is constant, then L(f + g) = L(f) + L(g) L(cf) = cL(f) That L has these properties follows readily from the corresponding properties for the derivative operators D, D2, Dn. Of course, not all differential equations are linear. Many important differential equations, such as dy + y2 = 0 dx are nonlinear. Second-Order Linear Equations A second-order linear differential equay + a1(x)y + a2(x)y = k(x) The theory of nonlinear differential equations is both complicated and fascinating, but best left for more advanced courses.ħ76 Chapter 15 Differential Equationstion has the form The presence of the exponent 2 on y is enough to spoil the linearity, as you may check. In this section, we make two simplifying assumptions: (1) a1(x) and a2(x) are constants, and (2) k(x) is identically zero. Thus, our initial task is to solve y + a1 y + a2 y = 0 A differential equation for which k(x) = 0 is said to be homogeneous.
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