MATH 1B Lecture Notes - Lecture 29: Integrating Factor
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Math 1b: calculus - lecture 29: variation of parameters. Q: consider (d2y/dx2) + y = tan(x), where 0 < x < pi/2. A: note that the corresponding homogeneous equation (d2y/dx2) + y = 0 has general solution y = To illustrate the philosophy of variation of parameters, let"s try to find a solution to * of the form y. [another way to think about it is u(x) = y/cos(x). ] So (dy/dx) = u"(x)cos(x) - u(x)sin(x). (d 2y/dx2) = u""(x)cos(x) - u"(x)sin(x) - u"(x)sin(x) - u(x)cos(x) For this to be a solution to *, we require u""(x)cos(x) - 2u"(x)sin(x) - u(x)cos(x) = tan(x). This is a 1st order differential equation in u"(x). The integrating factor is e -2tan(x)dx = e-2(ln|sec(x)|) = eln(cos(x)^2) = cos2(x). So we get: v"(x)cos2(x) - 2sin(x)cos(x)v(x) = sin(x). 2(x) = -cos(x) + c, where c is a constant. v"(x) - 2tan(x) = (sin(x)/cos. This is, in fact, the general solution the original equation, *.
