Higher Order Linear Differential Equations
Ordinary Differential Equations
Dr. Marco A Roque Sol
12/01/2015
Dr. Marco A Roque Sol
Ordinary Differential Equations
Higher Order Linear Differential Equations
General Theory of nth Order Linear Equations
Homogeneous Equations with Constant Coefficients
The Method of Undetermined Coefficients
The Method of Variation of Parameters
General Theory of nth Order Linear Equations
Example 4.2
Determine whether the functions f1 (t) = 1, f2 (t) = 2 + t,
f3 (t) = 3 − t 2 , and f4 (t) = 4t + t 2 are linearly independent or
dependent on the interval I : −∞ < t < ∞.
Solution
Form the linear combination
k1 f1 (t) + k2 f2 (t) + ... + k3 f3 (t) + k4 f4 (t) = k1 (1) + k2 (2 + t) + ...
... + k3 (3 − t 2 ) + k4 (4t − t 2 ) = 0
And differentiating the above formula three times we have
k2 − 2tk3 + k4 (4 − 2t) = 0
−2k3 − 2k4 = 0
0=0
Dr. Marco A Roque Sol
Ordinary Differential Equations
Higher Order Linear Differential Equations
General Theory of nth Order Linear Equations
Homogeneous Equations with Constant Coefficients
The Method of Undetermined Coefficients
The Method of Variation of Parameters
General Theory of nth Order Linear Equations
Thus, can show that the system has infinitely solutions and
therefore k1 , k2 , k3 , and k4 are not necesarily all of them equal to
zero. Hence, the set of functions is linearly dependent.
OBS
f4 (t) = 4t − t 2 = (3 − t 2 ) − 4(2 + t) + 5(1) = f3 (t) − 4f2 (t) + 5f1 (t)
therefore the set f1 (t) = 1, f2 (t) = 2 + t, f3 (t) = 3 − t 2 ,
f4 (t) = 4t + t 2 cannot be linearly indepent.
Dr. Marco A Roque Sol
Ordinary Differential Equations
Higher Order Linear Differential Equations
General Theory of nth Order Linear Equations
Homogeneous Equations with Constant Coefficients
The Method of Undetermined Coefficients
The Method of Variation of Parameters
General Theory of nth Order Linear Equations
Theorem 4.3
If y1 (t), ..., yn (t) is a fundamental set of solutions of the equation
L[y ](t) =
d ny
d n−1 y
dy
+
p
(t)
+ ... + pn−1
+ pn (t)y = 0
1
n
n−1
dt
dt
dt
on an interval I, then y1 (t), ..., yn (t) are linearly independent on I.
Conversely, if y1 (t), ..., yn (t) are linearly independent solutions of
the above equation on I, then they form a fundamental set of
solutions on I.
Dr. Marco A Roque Sol
Ordinary Differential Equations
Higher Order Linear Differential Equations
General Theory of nth Order Linear Equations
Homogeneous Equations with Constant Coefficients
The Method of Undetermined Coefficients
The Method of Variation of Parameters
General Theory of nth Order Linear Equations
The Nonhomogeneous Equation
Now consider the nonhomogeneous eq
L[y ](t) =
d ny
d n−1 y
dy
+ pn (t)y = g (t)
+
p
(t)
+ ... + pn−1
1
n
n−1
dt
dt
dt
If Y1 and Y2 are any two solutions of the above equation, then it
follows immediately from the linearity of the operator L that
L[Y1 − Y2 ](t) = L[Y1 ](t) − L[Y2 ](t) = g (t) − g (t) = 0
Dr. Marco A Roque Sol
Ordinary Differential Equations
Higher Order Linear Differential Equations
General Theory of nth Order Linear Equations
Homogeneous Equations with Constant Coefficients
The Method of Undetermined Coefficients
The Method of Variation of Parameters
General Theory of nth Order Linear Equations
Hence the difference of any two solutions of the nonhomogeneous
equation is a solution of the homogeneous equation. Since any
solution of the homogeneous equation can be expressed as a linear
combination of a fundamental set of solutions y1 , ..., yn , it follows
that any solution of nonhomogeneus equation can be written as
y (t) = c1 y1 (t) + c2 y2 (t) + ... + cn yn (t) + Y (t)
where Y is some particular solution of the nonhomogeneous
equation. The above linear combination is called the general
solution of the nonhomogeneous equation.
Dr. Marco A Roque Sol
Ordinary Differential Equations
Higher Order Linear Differential Equations
General Theory of nth Order Linear Equations
Homogeneous Equations with Constant Coefficients
The Method of Undetermined Coefficients
The Method of Variation of Parameters
Homogeneous Equations with Constant Coefficients
Let’s take the nth order linear homogeneous differential equation
d n−1 y
dy
d ny
+
a
+ ... + an−1
+ an y = 0
1
n
n−1
dt
dt
dt
where a0 , a1 , ..., an are real constants and a0 6= 0. Again we
proposed that y = e rt is a solution of the above equation. As a
matter of fact,
L[y ](t) = a0
L[e rt ] = e rt a0 r n + a1 r n−1 + ... + an−1 r + an = e rt Z (r )
for all r, where Z (r ) = a0 r n + a1 r n−1 + ... + an−1 r + an . For those
values of r for which Z (r ) = 0, it follows that L[e rt ] = 0 and
y = e rt is a solution of homogeneous equation. The polynomial
Z (r ) is called the characteristic polynomial , and the equation
Z (r ) = 0 is the characteristic equation .
Dr. Marco A Roque Sol
Ordinary Differential Equations
Higher Order Linear Differential Equations
General Theory of nth Order Linear Equations
Homogeneous Equations with Constant Coefficients
The Method of Undetermined Coefficients
The Method of Variation of Parameters
Homogeneous Equations with Constant Coefficients
Since a0 6= 0, we know that Z (r ) is a polynomial of degree n and
therefore as n zeros, say, r1 , r2 , ..., rn , some of which may be equal.
Hence we can write the characteristic polynomial in the form
(Fundamental Theorem of Algebra: Every non-constant
single-variable polynomial with complex coefficients has at least
one complex root. )
Z (r ) = a0 (r − r1 )(r − r2 ) . . . (r − rn ).
In general there are three cases, namely
1) Real and Different Roots.
If the roots of the characteristic equation are different, then we
have n distinct solutions e r1 t , e r2 t , ..., e rn t . These functions are
linearly independent, then the general solution is
Dr. Marco A Roque Sol
Ordinary Differential Equations
Higher Order Linear Differential Equations
General Theory of nth Order Linear Equations
Homogeneous Equations with Constant Coefficients
The Method of Undetermined Coefficients
The Method of Variation of Parameters
Homogeneous Equations with Constant Coefficients
y (t) = c1 e r1 t + c2 e r2 t + ... + cn e rn t
2) Complex Roots .
If the characteristic equation has complex roots, they must occur
in conjugate pairs, λ ± i µ, since the coefficients a0 , a1 , a2 , ..., an
are real numbers. Provided that none of the roots is repeated.
Now, just as for the second order equation, we can replace the
complex-valued solutions z1 = e (λ+i µ)t and z2 = e (λ−i µ)t by the
real-valued solutions
e λt cos(µt);
e λt sin(µt)
obtained as the linear cobinations 21 (z1 + z2 ) and
Dr. Marco A Roque Sol
1
2i (z1
Ordinary Differential Equations
− z2 )
Higher Order Linear Differential Equations
General Theory of nth Order Linear Equations
Homogeneous Equations with Constant Coefficients
The Method of Undetermined Coefficients
The Method of Variation of Parameters
Homogeneous Equations with Constant Coefficients
3) Repeated Roots.
Let’s assume that some of the roots are repeated. For an equation
of order n, if a root of Z (r ) = 0, say r = r1 , has multiplicity s (
that is, it is repeated s times) (where s ≤ n ), then
e r1 t , te r1 t + ... + t n−1 e r1 t
are corresponding solutions of the differential equation.
If a complex root λ + i µ is repeated s times, the complex
conjugate λ − i µ is also repeated s times. Corresponding to these
2s complex-valued solutions, we can find 2s real-valued linearly
independent solutions:
Dr. Marco A Roque Sol
Ordinary Differential Equations
General Theory of nth Order Linear Equations
Homogeneous Equations with Constant Coefficients
The Method of Undetermined Coefficients
The Method of Variation of Parameters
Higher Order Linear Differential Equations
Homogeneous Equations with Constant Coefficients
e λt cos(µt),
e λt sin(µt);
te λt cos(µt),
...; t n−1 e λt cos(µt),
te λt sin(µt); ...
t n−1 e λt sin(µt)
Hence the general solution of the homogeneous equation can
always be expressed as a linear combination of n real-valued
solutions.
Example 4.3
Find the general solution of the IVP
y (4) + y 000 − 7y 0 + 6y = 0; y (0) = 1,
y 0 (0) = 0,
y 00 (0) = −2,
y 000 (0) = −1
Dr. Marco A Roque Sol
Ordinary Differential Equations
Higher Order Linear Differential Equations
General Theory of nth Order Linear Equations
Homogeneous Equations with Constant Coefficients
The Method of Undetermined Coefficients
The Method of Variation of Parameters
Homogeneous Equations with Constant Coefficients
Solution
The characteristic equation is
r 4 + r 3 − 7r + 6 = 0
The roots of this equation are r1 = 1, r2 = −1, r3 = 2, and
r4 = −3. Therefore, the general solution is
y = c1 e t + c2 e −t + c3 e 2t + c4 e −3t
and aplying initial conditions we have
c1 + c2 + c3 + c4 = 1
c1 − c2 + 2c3 − 3c4 = 0
c1 + c2 + 4c3 + 9c4 = −2
c − c + 8c3 −
27c = 1
Ordinary 4Differential Equations
1 A Roque
2 Sol
Dr. Marco
Higher Order Linear Differential Equations
General Theory of nth Order Linear Equations
Homogeneous Equations with Constant Coefficients
The Method of Undetermined Coefficients
The Method of Variation of Parameters
Homogeneous Equations with Constant Coefficients
By solving this system of four linear algebraic equations, we find
that
c1 =
5
2
1
11
, c2 = , c3 = − , c4 = −
8
12
3
8
Thus the solution of the initial value problem is
y=
11 t
5
2
1
e + e −t − e 2t − e −3t
8
12
3
8
Dr. Marco A Roque Sol
Ordinary Differential Equations
Higher Order Linear Differential Equations
General Theory of nth Order Linear Equations
Homogeneous Equations with Constant Coefficients
The Method of Undetermined Coefficients
The Method of Variation of Parameters
Homogeneous Equations with Constant Coefficients
Example 4.4
Find the general solution of
y (4) + y = 0
Solution
The characteristic equation is
r 4 + 1 = 0 =⇒ r 4 = −1
In this way, we need to find the four roots of -1. Now −1, thought
of as a complex number, is −1 + 0i. It has magnitude 1 and polar
angle π (r = Re θi ). Thus
−1 = cos(π) + i sin(π) = e πi
Dr. Marco A Roque Sol
Ordinary Differential Equations
General Theory of nth Order Linear Equations
Homogeneous Equations with Constant Coefficients
The Method of Undetermined Coefficients
The Method of Variation of Parameters
Higher Order Linear Differential Equations
Homogeneous Equations with Constant Coefficients
Moreover, the angle is determined only up to a multiple of 2. Thus
−1 = cos(π + 2mπ) + i sin(π + 2mπ) = e (π+2mπ)i
where m is an integer. Thus
1/4
(−1)
=e
(π/4+2mπ/4)i
= cos
π 2mπ
+
4
4
+ i sin
π 2mπ
+
4
4
The four fourth roots of −1 are obtained by setting m = 0, 1, 2,
and 3;
Dr. Marco A Roque Sol
Ordinary Differential Equations
General Theory of nth Order Linear Equations
Homogeneous Equations with Constant Coefficients
The Method of Undetermined Coefficients
The Method of Variation of Parameters
Higher Order Linear Differential Equations
Homogeneous Equations with Constant Coefficients
m = 0 =⇒ cos
4
+ i sin
π 4
π π π
+
+ i sin
4
2
4
2
π
π
m = 2 =⇒ cos
+ π + i sin
+π
4
4
π 3π
π 3π
m = 3 =⇒ cos
+
+
+ i sin
4
2
4
2
m = 1 =⇒ cos
m = 4 =⇒ cos
π 4π
+
4
2
π
π +
+ i sin
Dr. Marco A Roque Sol
π 4π
+
4
2
= cos
Ordinary Differential Equations
π 4
+ i sin
π 4
!!!
Higher Order Linear Differential Equations
General Theory of nth Order Linear Equations
Homogeneous Equations with Constant Coefficients
The Method of Undetermined Coefficients
The Method of Variation of Parameters
Homogeneous Equations with Constant Coefficients
and the solutiones are
1 + i −1 + i −1 − i 1 − i
√ , √ , √ , √
2
2
2
2
The general solution is
t
t
√t
y (t) = e 2 c1 cos √
+ c2 sin √
+
2
2
t
t
− √t
e 2 c3 cos √
+ c4 sin √
2
2
Dr. Marco A Roque Sol
Ordinary Differential Equations
Higher Order Linear Differential Equations
General Theory of nth Order Linear Equations
Homogeneous Equations with Constant Coefficients
The Method of Undetermined Coefficients
The Method of Variation of Parameters
The Method of Undetermined Coefficients
We can find a particular solution YP of the nonhomogeneous nth
order linear equation with constant coefficients
L[y ](t) = a0
d ny
d n−1 y
dy
+
a
+ ... + an−1
+ an y = g (t)
1
n
n−1
dt
dt
dt
using The Method of Undetermined Coefficients, provided that
g (t) is of an appropriate form. We have to be careful when the
roots of the characteristic polynomial equation have multiplicity,
because now this could be greater than 2.
Example 4.5
Find the general solution of
y 000 − 3y 00 + 3y 0 − y = 4e t
Dr. Marco A Roque Sol
Ordinary Differential Equations
Higher Order Linear Differential Equations
General Theory of nth Order Linear Equations
Homogeneous Equations with Constant Coefficients
The Method of Undetermined Coefficients
The Method of Variation of Parameters
The Method of Undetermined Coefficients
Solution
The characteristic polynomial is
r 3 − 3r 2 + 3r − 1 = (r − 1)3
so the general solution of the homogeneous equation is
yc (t) = c1 e t + c2 te t + c3 t 2 e t
To find a particular solution YP (t) of the nonhomogeneous
equation we start by assuming that YP (t) = Ae t .
Dr. Marco A Roque Sol
Ordinary Differential Equations
Higher Order Linear Differential Equations
General Theory of nth Order Linear Equations
Homogeneous Equations with Constant Coefficients
The Method of Undetermined Coefficients
The Method of Variation of Parameters
The Method of Undetermined Coefficients
However, since e t , te t , and t 2 e t are all solutions of the
homogeneous equation, we must multiply this initial choice by t 3 .
Thus our final assumption is
YP (t) = At 3 e t =⇒ 6Ae t = 4e t =⇒ A =
Thus, the general solution is
2
yc (t) = c1 e t + c2 te t + c3 t 2 e t + t 3 e t
3
Dr. Marco A Roque Sol
Ordinary Differential Equations
2
3
Higher Order Linear Differential Equations
General Theory of nth Order Linear Equations
Homogeneous Equations with Constant Coefficients
The Method of Undetermined Coefficients
The Method of Variation of Parameters
The Method of Undetermined Coefficients
Example 4.6
Find a particular solution of
y 000 − 4y 0 = t + 3cos(t) + e −2t
Solution
The characteristic equation is
r 3 − 4r = 0
the roots are r = 0, ±2 and the general solution of the
homogeneous equation is
y (t) = c1 + c2 e 2t + c3 e −2t
.
Dr. Marco A Roque Sol
Ordinary Differential Equations
Higher Order Linear Differential Equations
General Theory of nth Order Linear Equations
Homogeneous Equations with Constant Coefficients
The Method of Undetermined Coefficients
The Method of Variation of Parameters
The Method of Undetermined Coefficients
To find a particular solution YP (t) of the nonhomogeneous
equation we propose that
YP (t) = (A0 t + A1 )t + A3 cost(t) + A4 sin(t) + A5 e −2t t
The constants are determined by substituting into the differential
equation. They are
A0 = −1/8, A1 = 0, A3 = 0, A4 = −3/5,
and
A5 = 1/8
.
Hence a particular solution is
1
3
1
YP (t) = − t 3 − sin(t) + te −2t
8
5
8
Dr. Marco A Roque Sol
Ordinary Differential Equations
Higher Order Linear Differential Equations
General Theory of nth Order Linear Equations
Homogeneous Equations with Constant Coefficients
The Method of Undetermined Coefficients
The Method of Variation of Parameters
The Method of Variation of Parameters
The method of variation of parameters for determining a particular
solution of the nonhomogeneous nth order linear differential
equation
L[y ](t) =
d n−1 y
dy
d ny
+
p
(t)
+ ... + pn−1
+ pn (t)y = g (t)
1
n
n−1
dt
dt
dt
Suppose then that we know a fundamental set of solutions
y1 , y2 , ..., yn of the homogeneous equation. Then the general
solution of the homogeneous equation is
y (t) = c1 y1 (t) + c2 y2 (t) + ... + cn yn (t)
Dr. Marco A Roque Sol
Ordinary Differential Equations