Lecture Notes in Differential Equations - Bruce E. Shapiro

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for y once z is known.
Method of Reduction of Order
If one solution y 1 is known for the differential equation
y ′′ + p(t)y ′ + q(t)y = 0
then a second solution is given by
∫ W (y1 , y 2 ))(t)
y 2 (t) = y 1 (t)
y 1 (t) 2 dt
where the Wronskian is given by Abel’s formula
( ∫ )
W (y 1 , y 2 )(t) = Cexp − p(s)ds
Method of Variation of Parameters
To find a particular solution to
y ′′ + p(t)y ′ + q(t)y = r(t)
when a pair of linearly independent solutions to the homogeneous equation
are already known,
∫
y p = −y 1 (t)
t
y ′′ + p(t)y ′ + q(t)y = 0
∫
y 2 (s)r(s)
W (y 1 , y 2 )(s) ds + y 2(t)
t
y 1 (s)r(s)
W (y 1 , y 2 )(s) ds
Power Series Solution
To solve
y ′′ + p(t)y ′ + q(t)y = g(t)
expand y, p, q and g in power series about ordinary (non-singular) points
and determine the coefficients by applying linear independence to the powers
of t.