![]() 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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