Worked examples and exercises are in the textSTROUD
PROGRAMME 25
SECOND-ORDER DIFFERENTIAL
EQUATIONS
Worked examples and exercises are in the textSTROUD
Programme 25: Second-order differential equations
Introduction
Homogeneous equations
The auxiliary equation
Summary
Inhomogeneous equations
Worked examples and exercises are in the textSTROUD
Programme 25: Second-order differential equations
Introduction
Homogeneous equations
The auxiliary equation
Summary
Inhomogeneous equations
Worked examples and exercises are in the textSTROUD
Programme 25: Second-order differential equations
Introduction
For any three numbers a, b and c, the two numbers:
are solutions to the quadratic equation:
with the properties:
2 2
1 2
4 4 and
2 2
b b ac b b acm m
a a
2 0am bm c
1 2 1 2 and b c
m m m ma a
Worked examples and exercises are in the textSTROUD
Programme 25: Second-order differential equations
Introduction
The differential equation:
can be re-written to read:
that is:
2
20
d y dya b cy
dx dx
2
20 provided 0
d y b dy cy a
dx a dx a
2
20
d y b dy ca y
dx a dx a
Worked examples and exercises are in the textSTROUD
Programme 25: Second-order differential equations
Introduction
The differential equation can again be re-written as:
where:
2 2
1 2 1 22 2
1 2 1
2
0
d y b dy c d y dyy m m m m y
dx a dx a dx dxd dy dy
m y m m ydx dx dx
dzm z
dx
2 2
1 2 1
4 4, and
2 2
b b ac b b ac dym m z m y
a a dx
Worked examples and exercises are in the textSTROUD
Programme 25: Second-order differential equations
Introduction
The differential equation:
has solution:
This means that:
That is:
2 0dz
m zdx
2
1
m x
dyz m y
dx
Ce
2 : being the integration constantm xz Ce C
21
m xdym y Ce
dx
Worked examples and exercises are in the textSTROUD
Programme 25: Second-order differential equations
Introduction
The differential equation:
has solution:
where: and are constantsA B
1 2
1
1 2
1 2
: if
( ) : if
m x m x
m x
y Ae Be m m
A Bx e m m
21
m xdym y Ce
dx
Worked examples and exercises are in the textSTROUD
Programme 25: Second-order differential equations
Introduction
Homogeneous equations
The auxiliary equation
Summary
Inhomogeneous equations
Worked examples and exercises are in the textSTROUD
Programme 25: Second-order differential equations
Homogeneous equations
The differential equation:
Is a second-order, constant coefficient, linear, homogeneous differential equation. Its solution is found from the solutions to the auxiliary equation:
These are:
2
20
d y dya b cy
dx dx
2 0am bm c
2 2
1 2
4 4 and
2 2
b b ac b b acm m
a a
Worked examples and exercises are in the textSTROUD
Programme 25: Second-order differential equations
Introduction
Homogeneous equations
The auxiliary equation
Summary
Inhomogeneous equations
Worked examples and exercises are in the textSTROUD
Programme 25: Second-order differential equations
The auxiliary equation
Real and different roots
Real and equal roots
Complex roots
Worked examples and exercises are in the textSTROUD
Programme 25: Second-order differential equations
The auxiliary equation
Real and different roots
If the auxiliary equation:
with solution:
where:
then the solution to:
2 0am bm c
2 2
1 2
4 4 and
2 2
b b ac b b acm m
a a
1 2 1 2 and are real and m m m m
1 2
2
20 is m x m xd y dy
a b cy y Ae Bedx dx
Worked examples and exercises are in the textSTROUD
Programme 25: Second-order differential equations
The auxiliary equation
Real and equal roots
If the auxiliary equation:
with solution:
where:
then the solution to:
2 0am bm c
2 2
1 2
4 4 and
2 2
b b ac b b acm m
a a
1 2 1 2 and are real and m m m m
1
2
20 is ( ) m xd y dy
a b cy y A Bx edx dx
Worked examples and exercises are in the textSTROUD
Programme 25: Second-order differential equations
The auxiliary equation
Complex roots
If the auxiliary equation:
with solution:
where:
Then the solutions to the auxiliary equation are complex conjugates. That is:
2 0am bm c
2 2
1 2
4 4 and
2 2
b b ac b b acm m
a a
1 2 and are m m complex
1 2 and m j m j
Worked examples and exercises are in the textSTROUD
Programme 25: Second-order differential equations
The auxiliary equation
Complex roots
Complex roots to the auxiliary equation:
means that the solution of the differential equation:
is of the form:
2 0am bm c
2
20
d y dya b cy
dx dx
( ) ( )j x j x
x j x j x
y Ae Be
e Ae Be
Worked examples and exercises are in the textSTROUD
Programme 25: Second-order differential equations
The auxiliary equation
Complex roots
Since:
then:
The solution to the differential equation whose auxiliary equation has complex roots can be written as::
cos sin and cos sinj x j xe x j x e x j x
cos sinxy e C x D x
( )cos ( )sin
cos sin
j x j xAe Be A B x j A B x
C x D x
Worked examples and exercises are in the textSTROUD
Programme 25: Second-order differential equations
Introduction
Homogeneous equations
The auxiliary equation
Summary
Inhomogeneous equations
Worked examples and exercises are in the textSTROUD
Programme 25: Second-order differential equations
Summary
Differential equations of the form:
Auxiliary equation:
Roots real and different: Solution
Roots real and the same: Solution
Roots complex ( j): Solution
2
20 where , and are contants
d y dya b cy a b c
dx dx
21 20 with roots and am bm c m m
1 2m x m xy Ae Be
1( ) m xy A Bx e
cos sinxy e C x D x
Worked examples and exercises are in the textSTROUD
Programme 25: Second-order differential equations
Introduction
Homogeneous equations
The auxiliary equation
Summary
Inhomogeneous equations
Worked examples and exercises are in the textSTROUD
Programme 25: Second-order differential equations
Inhomogeneous equations
The second-order, constant coefficient, linear, inhomogeneous differentialequation is an equation of the type:
The solution is in two parts y1 + y2:
(a) part 1, y1 is the solution to the homogeneous equation and is called the complementary function which is the solution to the homogeneous equation
(b) part 2, y2 is called the particular integral.
2
2( )
d y dya b cy f x
dx dx
Worked examples and exercises are in the textSTROUD
Programme 25: Second-order differential equations
Inhomogeneous equations
Complementary function
Example, to solve:
(a) Complementary function
Auxiliary equation: m2 – 5m + 6 = 0 solution m = 2, 3
Complementary function y1 = Ae2x + Be3x where:
22
25 6
d y dyy x
dx dx
21 1
125 6 0
d y dyy
dx dx
Worked examples and exercises are in the textSTROUD
Programme 25: Second-order differential equations
Inhomogeneous equations
Particular integral
(b) Particular integral
Assume a form for y2 as y2 = Cx2 + Dx + E then substitution in:
gives:
yielding:
so that:
222 2
225 6
d y dyy x
dx dx
2 26 (6 10 ) (2 5 6 ) 0 0Cx D C x C D E x x
1/ 6 : 5 /18 : 19 /108C D E
2
2
5 19
6 18 108
x xy
Worked examples and exercises are in the textSTROUD
Programme 25: Second-order differential equations
Inhomogeneous equations
Complete solution
(c) The complete solution to:
consists of:
complementary function + particular integralThat is:
22 3
1 2
5 19
6 18 108x x x x
y y y Ae Be
22
25 6
d y dyy x
dx dx
Worked examples and exercises are in the textSTROUD
Programme 25: Second-order differential equations
Inhomogeneous equations
Particular integrals
The general form assumed for the particular integral depends upon the form of the right-hand side of the inhomogeneous equation. The following table can be used as a guide:
2 2
( ) Assume
sin or cos sin cos
sinh or cosh sinh coshkx kx
f x y
k C
kx Cx D
kx Cx Dx E
k x k x C x D x
k x k x C x D x
e Ce
Worked examples and exercises are in the textSTROUD
Learning outcomes
Use the auxiliary equation to solve certain second-order homogeneous equations
Use the complementary function and the particular integral to solve certain second-order inhomogeneous equations
Programme 25: Second-order differential equations