MATH 2574H Lecture : Lecture1292014.pdf
Document Summary
F(t, y, dy/dt, d2y/dt2, , dny/dtn) = 0 te(a,b) Initial value problems: given differential eqn function and some initial value constants: y(a) = c1, y"(a) = c2; Boundary value problem (bvp); y(b) = c1, y(a) = c2; Linear diff eqn (ode) can be written in the form: a0(t)y(m) + a1(t)y(n-1) + an(t)y = g(t) where y(n) = dny/dtn; First order linear ode (highest derivative is 1st) a0(t)(dy/dt) + a1(t)y = g(t) -->can divide out by a(t) dy/dx + p(x)y = q(x) --> general form. Chose f(x) = integrate[p(x)dx] -->can also add any integration factor c (so not unique function) but derivative always is the same y(x) = e-integrate[p(x)dx][integrate[q(x)eintegrate[p(x)dx]dx + c] ex: y" + 2xy = 0 --> find the general solution. P(x) = 2x --> f(x) = integrate[2xdx] = x2; ef(x) = ex^2 --> aka integrating factor y"ex^2 + 2xex^2y = 0 (d/dx)(y"ex^2) = 0 y"ex^2 = c. Integrating factor: ef(x) = exp[(1/2)ln(x2 - 1)] = (x2 - 1)1/2.
