PPT: Higher Order Linear Differential Equations with Constant Coefficients | Engineering Mathematics - Civil Engineering (CE) PDF Download

 Page 1


Introduction
? A differential equation is an equation which contains 
the derivatives of a variable, such as the equation
?? ?? 2
?? ????
2
+ ?? ????
????
+ ???? = ?? Here x is the variable and a, b, c and d are constants. 
Page 2


Introduction
? A differential equation is an equation which contains 
the derivatives of a variable, such as the equation
?? ?? 2
?? ????
2
+ ?? ????
????
+ ???? = ?? Here x is the variable and a, b, c and d are constants. 
Types of Differential Equations
? Homogeneous DE
? Non Homogeneous DE
0 ) ( ) ( ) ( ) (
0 1
1
1
1
? ? ? ? ?
?
?
?
y x a
dx
dy
x a
dx
y d
x a
dx
y d
x a
n
n
n
n
n
n
?
) ( ) ( ) ( ) ( ) (
0 1
1
1
1
x g y x a
dx
dy
x a
dx
y d
x a
dx
y d
x a
n
n
n
n
n
n
? ? ? ? ?
?
?
?
?
Page 3


Introduction
? A differential equation is an equation which contains 
the derivatives of a variable, such as the equation
?? ?? 2
?? ????
2
+ ?? ????
????
+ ???? = ?? Here x is the variable and a, b, c and d are constants. 
Types of Differential Equations
? Homogeneous DE
? Non Homogeneous DE
0 ) ( ) ( ) ( ) (
0 1
1
1
1
? ? ? ? ?
?
?
?
y x a
dx
dy
x a
dx
y d
x a
dx
y d
x a
n
n
n
n
n
n
?
) ( ) ( ) ( ) ( ) (
0 1
1
1
1
x g y x a
dx
dy
x a
dx
y d
x a
dx
y d
x a
n
n
n
n
n
n
? ? ? ? ?
?
?
?
?
Second Order Homogeneous DE
? A linear second order homogeneous differential 
equation involves terms up to the second derivative 
of a function. For the case of constant multipliers, 
The equation is of the form
? and can be solved by the substitution
Page 4


Introduction
? A differential equation is an equation which contains 
the derivatives of a variable, such as the equation
?? ?? 2
?? ????
2
+ ?? ????
????
+ ???? = ?? Here x is the variable and a, b, c and d are constants. 
Types of Differential Equations
? Homogeneous DE
? Non Homogeneous DE
0 ) ( ) ( ) ( ) (
0 1
1
1
1
? ? ? ? ?
?
?
?
y x a
dx
dy
x a
dx
y d
x a
dx
y d
x a
n
n
n
n
n
n
?
) ( ) ( ) ( ) ( ) (
0 1
1
1
1
x g y x a
dx
dy
x a
dx
y d
x a
dx
y d
x a
n
n
n
n
n
n
? ? ? ? ?
?
?
?
?
Second Order Homogeneous DE
? A linear second order homogeneous differential 
equation involves terms up to the second derivative 
of a function. For the case of constant multipliers, 
The equation is of the form
? and can be solved by the substitution
Solution
The solution which fits a specific physical situation is 
obtained by substituting the solution into the equation and 
evaluating the various constants by forcing the solution to 
fit the physical boundary conditions of the problem at 
hand. Substituting gives
which leads to a variety of solutions, depending on the 
values of a and b. In physical problems, the boundary 
conditions determine the values of a and b, and the 
solution to the quadratic equation for ? reveals the nature 
of the solution. 
Page 5


Introduction
? A differential equation is an equation which contains 
the derivatives of a variable, such as the equation
?? ?? 2
?? ????
2
+ ?? ????
????
+ ???? = ?? Here x is the variable and a, b, c and d are constants. 
Types of Differential Equations
? Homogeneous DE
? Non Homogeneous DE
0 ) ( ) ( ) ( ) (
0 1
1
1
1
? ? ? ? ?
?
?
?
y x a
dx
dy
x a
dx
y d
x a
dx
y d
x a
n
n
n
n
n
n
?
) ( ) ( ) ( ) ( ) (
0 1
1
1
1
x g y x a
dx
dy
x a
dx
y d
x a
dx
y d
x a
n
n
n
n
n
n
? ? ? ? ?
?
?
?
?
Second Order Homogeneous DE
? A linear second order homogeneous differential 
equation involves terms up to the second derivative 
of a function. For the case of constant multipliers, 
The equation is of the form
? and can be solved by the substitution
Solution
The solution which fits a specific physical situation is 
obtained by substituting the solution into the equation and 
evaluating the various constants by forcing the solution to 
fit the physical boundary conditions of the problem at 
hand. Substituting gives
which leads to a variety of solutions, depending on the 
values of a and b. In physical problems, the boundary 
conditions determine the values of a and b, and the 
solution to the quadratic equation for ? reveals the nature 
of the solution. 
Case I: Two real roots
? For values of a and b such that
• there are two real roots m
1
and m
2
which lead to a general 
solution of the form
2
1
12
()
n
m x m x
mx
n
y x ce c e c e ? ? ? ? K