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2332 Tutorial Sheet 11 Useful facts: • To solve the equation ay 00 +by 0 +cy = 0, with a, b and c constants, use an exponential substitution y = exp(λx) and solve for λ. If this only gives one solution, then y = x exp(λx) is also a solution. • The general solution of the equation ay 00 + by 0 + cy = d exp (µx), with a, b, c and d constants is of the form y = yp +yc where yc is the general solution to ay 00 +by 0 +cy = 0. To find yp , substitute yp = C exp µx and solve for C. This doesn’t work is expµx solves ay 00 + by 0 + cy = 0, in which case substitute yp = Cx exp µx, this in turn won’t work if x exp µx also solves ay 00 + by 0 + cy = 0, in which case substitute yp = Cx2 exp µx. • To solve ay 00 + by 0 + cy = f (x) where f (x) isn’t an exponetial, then write f (x) in terms of exponentials, using Fourier methods if needed. For example, say f (x) = ∞ X cn einx n=−∞ then find a particular solution yn to the equation ay 00 + by 0 + cy = cn einx for all n, the general solution to ay 00 + by 0 + cy = f (x) is then y = yc + ∞ X yn n=−∞ where, again, yc solves the corresponding homogeneous equation. 1 Sinéad Ryan, see http://www.maths.tcd.ie/˜ryan/231.html 1 Questions 1. Obtain a general solution to (a) y 0 − 3y = e−x (b) y 0 + y cot x = cos x (c) (x + 1)y 0 + y = (x + 1)2 2. Obtain the general solutions of the following ODEs: (a) y 00 + 5y 0 + 6y = 0 (b) y 00 − 2y 0 + y = 0 (c) y 00 + 2y 0 + y = sin 2x (d) y 00 + 2y 0 + y = x2 3. Obtain the general solution of the ODE y 00 (x) + 3y 0 (x) + 2y(x) = f (x) where f is the periodic function defined 0 1 f (x) = 0 by −π < x < −a −a < x < a a<x<π where a ∈ (0, π) is a constant and f (x + 2π) = f (x). 2