MAP 2302
Differential Equations
Summary 9:Higher Order Differential Equations
Sanchez
DE equations reducible in order
I. Case I. Equations of the form y (n)  F( x, y (n 1) ) can be reduced to first order by the
transforma tion w  y (n 1)
II . Case II . A second  order differenti al equations of the form y   F( y , y  )
dw
can be reduced to first order by the transforma tion w  y  and y   w
dy
dy
dy  dw dw dy dw
dw
Note : w  y  
 y  




 y 
w
dx
dx
dx
dy dx
dy
dy
III. Reduction of order- D’Alembert’s Procedure.
If y1 is a solution of the related homogeneous equation of a given differential equation,
Then the substitution y=vy1 and then v’=w produces a linear differential equation of order n-1.
Proof: (for n=2)
1. Let y   Py   Qy  R be a differenti al equation
2. Let y   Py   Qy  0 be the related hom ogeneous differenti al equation
3. y1 is a solution of y   Py   Qy  0  y1  Py1  Qy 1  0
4. y  vy1  y   v y1  vy1 and y   v y1  v y1  v y1  vy1
y   v y1  2 v y1  vy1
5. y   Py   Qy  R  v y1  2 v y1  vy1  Pv y1  vy1   Qvy 1  R
 v y1  v ( 2y1  Py 1 )  vy1  Py1  Qy 1   R
 v y1  v ( 2y1  Py 1 )  v0   R
Let w  v  and w   v   y1w   ( 2y1  Py 1 )w  R
IV. Abel’s Formula:
If y1 is a solution of a second order hom ogeneous linear differenti al equation ,
y   P( x )y   Q( x )y  0,
then y 2  y1 
 p( x )dx
e 
y1 2
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dx
Linearly independent solutions of a differential equation
Wronskian of a set of functions.
The Wronskian of the set of functions f1(x), f2(x), f2(x),…, fn(x) is defined by the determinant
f1 ( x)
f 2 ( x)
...
f n ( x)
f1 ( x)
f 2 ( x)
...
f n ( x
.
.
.
.
W  (f1 , f 2 , . . ., f n ) 
provided the functions
.
.
.
.
.
.
.
.
(n 1)
f1
(n 1)
( x) f 2
( x)
(n 1)
. . . f1
( x)
have n  1 derivative s.
Test for independence of a set of functions:
The set of functions f1 , f 2 , . . ., f n  is linearly independen t if the Wronskian W(f1 , f 2 , . . ., f n )  0
for every x in the int erval .
Homogeneous Differential Equations of order n
Linear Differential Operator of order n.
If Dk y 
dk y
k
, then P  an Dn  an 1Dn 1  an  2 Dn  2  . . .  a 2 D2  a1D  a 0
dx
is called a Linear Differential Operator of order n.
Properties of the solutions of an nth order homogeneous linear differential equation:
1) Any linear combination of solutions is also a solution. That is,
if y1 and y 2 are solutions and c1 anf c 2 are any cons tan ts, then c1y1  c 2 y 2 is also a solution
2) An nth order homogeneous linear differential equation has n linearly independent solutions:
y1 , y 2 , y 3 , . . , y n .
3). The general solution of the homogeneous linear differential equation is given by
y h  c1y1  c 2 y 2  . . .  c n y n .
4) If Yh is the homogeneous solution of a non-homogeneous differential equation and Yp is a
particular solution of the non-homogeneous differential equation, then
Y = Yh + Yp is the general solution of the non-homogeneous differential equation.
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5) Theorem.
If y  e ax is a solution of the hom ogeneous differenti al equation with constant coefficients
an
dn y
dx
n
 a n 1
d n  1y
dx
n 1
 ...  a2
d2y
dx
2
 a1
dy
 a o y  0, then " a" is a solution of
dx
the polynomial function a n x n  a n 1x n 1  . . .  a 2 x 2  a1x  a 0  0
Definition : the polynomial function a n x n  a n 1x n 1  . . .  a 2 x 2  a1x  a 0  0
is called the Characteri stic Equation of the linear hom ogeneous differenti al
equation with constant coefficients. The solutions are called the Eigenvalue s of the DE.
Definition : an Dn  an 1Dn 1  . . .  a 2 D2  a1D  a 0  0
is called the differenti al operator of the linear hom ogeneous differenti al equation
with constant coefficients .
Theorem:
If y  e ax is a solution of the hom ogeneous differenti al equation with constant coefficients
an
dn y
dx
n
 a n 1
d n  1y
dx
n 1
 ...  a2
d2y
dx
2
 a1
dy
 a o y  0, then " a" is a solution of
dx
the polynomial function a n x n  a n 1x n 1  . . .  a 2 x 2  a1x  a 0  0
Definition : the polynomial function a n x n  a n 1x n 1  . . .  a 2 x 2  a1x  a 0  0
is called the Characteri stic Equation of the linear hom ogeneous differenti al
equation with constant coefficients . The solutions are called the Eigenvalue s of the DE.
Definition : a n D n  a n 1D n 1  . . .  a 2 D 2  a1D  a 0  0
is called the differenti al operator of the linear hom ogeneous differenti al equation
with constant coefficients .
I. Theorem- If m is a double root of the characteristic polynomial of a Homogeneous Linear
Differential Equation with constant coefficients then
y1  e mx and y 2  xemx are linearly independen t solutions of the differenti al equation .
Theorem . If m is a multiple root (k times ) of the characteri stic equation of a
homogeneous linear differential equation with constant coefficients, then
y  (c1  c 2 x  c3x 2  . . .  ck xk 1 )emx is a hom ogeneous solution of the DE.
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Theorem : If m1     i and m 2     i are conjugate complex roots of the characteri stic
polynomial of a linear hom ogeneous DE with cons tan t coefficients, then
y  e x (c1 cos  x  c 2 sin  x) is a solution of the DE.
Definition: Boundary Value Problems (BVP): a linear differential equation of order two or greater in
which the dependent variable y or its derivatives are specified at different points.
A boundary value problem can have many, one, or no solutions.
Properties of linear differential operators.
1. If P and Q are linear differential operators then PQy=P(Qy)
2. If P and Q are linear differential operators with constant coefficients then PQ=QP
3. Dm Dn  Dm  n
4. Dk e ax  a k e ax
5. f ( D)e ax  e ax f (a)
6a. First Shifting Formula
 
D e ax y  e ax D  a  y
Note:
 
6b. e ax Dy  ( D  a) e ax y
7a. Second Shifting Formula
 
D n e ax y  e ax D  a n y
 
7b. Note : e ax Dn y  ( D  a)n e ax y
 
8a. f (D) e ax y  e ax f ( D  a)y 
 
8) e ax f ( D)y   f ( D  a) e ax y
 
9. f D  a  e ax y  e ax f D y
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Undeterminate Coefficients
II. Un-determinate coefficients. This method can be applied if
1) The coefficients of the DE are constants
2) g(x) is a constant, a polynomial , an exponential function, a sine or a cosine function, or a finite sum
or product of these functions.
Solution:
1. Find the homogeneous solution Yh
2. The form of the particular solution Yp is a linear combination of all linearly independent functions
that are generated by repeated differentiations of g(x)
3. If any Ypi contains terms that duplicate terms in the homogeneou s solution Yh ,
then Ypi must be multiplied by x n , where n is the smallest positive integer th at eliminates
that duplicatio n.
4. Substitute the Yp obtained in the differential equation
5. Use the method of un-determinate coefficients to solve for the constants.
6. The non-homogeneous solution of the differential equation is y=Yh+Yp.
Annihilators
Annihilators. An annihilator of a function y=f(t) is a linear differential P(D) that satisfies the condition
P(D)[f(t)] =0. This is the same as saying that f(t) is a solution of the homogenous differential equation
P(D)f(t)=0.
Finding the general form of the particular solution of a non-homogeneous linear differential equation
by using annihilators.
Step 1. Express the DE in linear differential form, that is, L(y)=g(x)
Step 2. Find the homogeneous solution (complementary solution) of the differential equation, that is
find the general solution of L(y)=0
Step 3.. Find an annihilator L1for g(x), that is L1(g(x)=0
Step 4. Operate on both sides of the non-homogeneous equation with the annihilator L1, that is,
L1 L(y )  L1 (g(x)  0
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Step 5. Find the homogeneou s solution (complemen tary solution) of the differenti al
equation L1 L( y )  0
Step 6. The general for the particular solution of L( y )  g( x ) is given by
e lim inating from the solution of L1 L( y )  0, the terms which belong to the
solution of L( y )  0
Variation of parameters technique.
Technique:
Step 1. Let y h  c1y1  c 2 y 2 be the hom ogeneous solution
Step 2. Assume y p  v1y1  v 2 y 2 where v1 and v 2 are functions and
v1 y1  v 2 y 2  0
Step 3. Solve the system for v1 , v 2
 v1 y1  v 2 y 2  0

 v1 y1  v 2 y 2  g( x )
Step 4. Integrate each of v1 , v 2 , n to find v1 , v 2
Step 5. Find y p  v1y1  v 2 y 2
Step 6. Find y  y h  y p
Technique: The above technique can be generalized to a differential of order n.
Step 1. Let y h  c1 y1  c2 y 2  ...  cn y n be the hom ogeneous solution
Step 2. Assume y p  v1 y1  v2 y 2  ...  vn y n where vi are functions
Step 3. Solve the system for v1, v2 , ..., vn
v1 y1  v2 y 2  ...  vn y n  0


v1 y1  v2 y 2  ...  vn y n  0


v1 y1  v2 y 2  ...  vn y n  0




v y ( n 1)1  v y ( n 1) 2  ...  v y ( n 1) n  g ( x)
2
n
 1
an

Step 4. Integrate each of v1, v2 , ..., vn to find v1 , v2 , ..., vn
Step 5. Find y p  v1 y1  v2 y 2  ...  vn y n
Step 6. Find y  y h  y p
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Cauchy-Euler Differential Equation :
Definition:A DE of the form
an x
n
d (n ) y
 a n1x
dx n 
n 1
d (n1) y
2
d2y
dy
 . . .  a2x
 a1x
 a 0 y  g( x )
dx
dx n1
dx 2
is called a Cauchy-Euler Differential Equation
Theorem:
If x  0, the substituti on x  e t , t  ln x, transforms the equation to a linear
DE of order n with constant coefficients .
If x  0, the substituti on  x  e t , t  ln(  x) transforms the equation to a linear
DE of order n with constant coefficients .
Technique:
m
Step 1. Write the characteri stic equation  a k m (m  1)(m  2) . . . m  k  1  0
k 0
and solve for m.
Step 2. Follow the procedures used for Linear equations of order n with constant coefficients.
Step 3. Write the answer in the form y=h(t).
Step 4. If y  f(x), let x  e t and t  ln x
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