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.
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Sinéad Ryan, see http://www.maths.tcd.ie/˜ryan/231.html
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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
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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).
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