2373 Math Week 8 Toews
1
Reduction of Order
y (x) + p(x)y (x) + q(x)y(x) = f (x).
Consider the general second order linear non-homogenous equation (1) (2)
If q 0, this becomes
y (x) + p(x)y (x) = f (x),
for which we can make the substitution u = y to get u (x) + p(x)u(x) = f (x). (3)
Equation (3) is rst order linear in u, and can be solved. Once u is known, y can found as the integral of u. In the general case, this substitution will not work. However, if we happen to know one solution y1 to the homogenous equation (1), a second solution can be assumed to have the form y2 (x) = u(x)y1 (x). (4)
Expanding y2 and y2 using the product rule, substituting in (1), and using the fact that y1 is a solution of the homogenous equation, we get
u + u p(x) + 2
f y1 = . y1 y1
(5)
Equation (5) is rst order linear, and can be solved for u. Once u is known, y2 is given by (4).
2
Resonance
The method of undetermined coecients amounted to guessing an appropriate form of the particular solution for the equation y + ay + by = f (6)
where f is either trigonometric, or polynomial, exponential (or some combination of these.) The notes for Week 6 outline the appropriate guesses whenever f is not also a solution to the homogenous equation. If f is a solution to the 1
homogenous equation, the usual guess should be multiplied by a factor of x. A proof of this fact can be developed using the method of Reduction of Order. Recall that spring systems and LRC problems involve an equation of the form (6), where the coecients are positive. If the spring system has no damping or the LRC system has no resistence, then the second term in (6) is zero and the solution to the homogenous equation is a f...