MATH 1B Lecture 27: Even More Nonhomogeneous 2nd Order Differential Equations
Document Summary
Math 1b: calculus - lecture 27: even more nonhomogeneous 2nd order differential. Q: find the general solution to (d2y/dx2) + 3(dy/dx) + 2y = x+1. A: we have (d2y/dx2) + 3(dy/dx) + 2y = x+1. The corresponding homogeneous equation is: (d2y/dx2) + 3(dy/dx) + 2y = 0. This has an auxiliary equation r2 + 3r + 2 = 0. So, the auxiliary equation has 2 real solutions, r = -1 and r = -2. So, the homogeneous equation has a general solution y = ae-x + be-2x, where a and b are constants. To find a particular solution to *, we will try y=c, x+c2. So (dy/dx) = c1, and (d2y/dx2) = 0. This gives (d2y/dx2) + 3(dy/dx) + 2y (r+1)(r+2) = 0. We want 2c1 = 1 and 3c1 + 2c2 = 1. So, c1 = and c2 = (1 - 3c1)/2. So, we have y = x - as our particular solution to *.
