18.03 Lecture Notes - Lecture 9: Integrating Factor, Logistic Function, Smart People
Document Summary
[3] potential blow-up of solutions to a nonlinear equation. Review: nonlinear vs linear y"(t) = f(t,y(t)) vs r(t) x"(t) + p(t) x(t) = q(t) This is in the form of a debate, between linn e. r. (on the right) and chao s. (on the left). Sometimes you can just *see* this: t^2 x" + 2t x = (d/dt)(t^2 x) for example. If we are in *reduced* standard form, so r = 1 , then this can be done systematically: We seek u(t) such that u (x" + px) = (d/dt) (ux) i. e. pu = u" : separable, with solution u = e^{\int p(t) dt} (any constant of integration will do here). Then integrate both sides of (d/dt) (ux) = uq : x = u^{-t} int uq dt. The constant of integration is in this integral, so the general solution has the form x = x_p + c u^{-1} .
