Difference Equations to Differential Equations (50).pdf

Section
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 First Order Linear
 ¢¡£¡¥¤§¦¨¤§© £
 £© Differential Equations
We will now consider closed form solutions for another important class of differential
equations. A differential equation
x˙= f(x, t)
with x(t0) = x0 is called a linear equation if
f(x, t) = p(t)x + q(t) ()
for some functions p and q which depend only on t. We will assume that both p and q are
continuous functions. Note that under certain circumstances, such as q(t) = 0 for all t, a
linear equation is also separable. The solution of such equations is based on the following
observation: If we let
t
P (t) = Z p(s)ds, ()
t0
then
d
xe−P (t) = −xp(t)e−P (t) +xe ˙−P (t) = e−P (t)(x ˙− p(t)x). ()
dt


Now we wantx ˙= p(t)x + q(t), that is,x ˙− p(t)x = q(t), so we are looking for a function x
such that
d
xe−P (t) = q(t)e−P (t). ()
dt


Integrating () from t0 to t (using u for our variable of integration), we have
t t
d −P (u) −P (u)
Z x(u)e du = Z q(u)e du. ()
t0 du

t0
Now
t t
d −P (u) −P (u)
Z x(u)e du = x(u)e
du t
t0

 0
 ()
−P (t) −P (t0)
= x(t)e − x(t0)e
−P (t)
= x(t)e − x0
since P (t0) = 0 and x(t0) = x0. Hence we want
t
−P (t) −P (u)
x(t)e − x0 = Z q(u)e du. ()
t0
1
2 First Order Linear Differential Equations Section