Chapter 21
Linear, First-Order Differential Equations
There are two types of differential equations:
1. Autonomous: Differential equation which is not an explicit function of time t.
2. Non-Autonomous: Differential equation
which is an explicit function of time t.
The general form of the linear, autonomous, first-order differential equation is
ẏ + ay = b
(21.1)
where a and b are known constants
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In order to solve (21.1), we divide the
differential equation into two parts:
1. Homogeneous part (or form) which is obtained by setting b = 0, i.e. the homogeneous form of the differential equation
is
ẏ + ay = 0, ∀a 6= 0.
(21.2)
The solution to the homogeneous part is
known as homogenous solution denoted by yh .
2. Steady-state part, which is obtained by
setting ẏ = 0
ay = b.
(21.3)
The solution to (21.3) is known as particular solution or steady state solution. It is a value of y denoted by y or
yp , at which y is stationary or does not
change over time.
2
The general solution to the above differential
equation given in (21.1) is sum of homogeneous and particular (or steady state solution):
y = yh + yp .
(21.4)
Theorem 21.1 The general solution to
the homogeneous form of the linear, autonomous, first order differential equation is
yh (t) = C exp−at
(21.5)
where C is some constant.
The particular or steady state solution of the linear, autonomous, first
order differential equation is
b
y ≡ yp = .
a
3
(21.6)
Theorem 21.3: The general solution to
the complete autonomous, linear, firstorder differential equation is
b
y(t) = C exp
+ .
(21.7)
a
Steady State and Convergence
−at
Question is: whether a variable y(t) reaches
its steady state value y as time passes by? If
it does then we say that variable y(t) converges to the steady-state equilibrium value.
Technically, we are looking at
−at
lim y(t) = lim [C exp
t→∞
t→∞
If limt→∞ y(t) = y =
achieved.
b
a,
4
b
+ ] =?.
a
(21.8)
then convergence is
Theorem 21.4: The solution to a linear, first-order differential equation, y(t),
converges to the steady state equilibrium, y = ab , iff the coefficient in the differential equation is positive: a > 0.
Initial Value Problem
Apart from general form of differential
equation, suppose that we are also given the
initial value for y, i.e., the value of y at t = t0 ,
where t0 is the initial value of t, then the solution to the differential equation must satisfy
the initial value. Let y0 be the initial value
of y then
b
y0 = C exp
+ .
(21.9)
a
The initial value allows us to pin down the
value of constant C, which is equal to
−at0
µ
¶
b
C = y0 −
expat0 .
a
5
(21.10)
Dynamics of National Debt
Question: Can a government run persistent
budget deficit? Can national debt grow forever without bankrupting the nation?
Answer depends on the growth rates of debt,
national income, and interest rate. Let
D(t) = Debt at time t
y(t) = National income at time t
r = Interest rate on debt
rD(t) = Interest payment at time t
D(0) = 0
y(0) = 1
Here bankruptcy means whether the ratio of interest payment to national income
will exceed one i.e. rD(t)
y(t) ≥ 1 at some point
in time.
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Suppose that time path of debt is given
by
Ḋ = by(t), b > 0
and the time path of income is given by
ẏ = gy(t), g > 0
Given the set up, we want to know:
1. The time path of
rD(t)
y(t) .
2. What is the steady-state value of
Does it exceed one ?
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rD(t)
y(t) ?
Non-autonomous Equations
The general form of the linear, first-order
differential equation is
ẏ + a(t)y = b(t)
(21.11)
where a(t) and b(t) are known, continuous functions of t.
In order to solve such equation, we make
use of integrating factor. We multiply both
sides of (21.11) by the integrating factor. This makes the differential equation amenable to
direct integration.
Theorem 21.6: The general form of
the integrating factor for the linear, firstorder differential equation is
expA(t)
R
where A(t) = a(t)dt.
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(21.12)
Theorem 21.5: The general solution to
any linear, first-order differential equation is
·Z
y(t) = exp−A(t)
where A(t) =
¸
expA(t) b(t)dt + C
R
(21.13)
a(t)dt.
Note that the non-autonomous differential equation usually does not have steadystate solution as its coefficients are time-varying or functions of time.
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