PPT: Method of Parameter Variations | Engineering Mathematics - Civil Engineering (CE) PDF Download

 Page 1


Method of Variation of Parameters
› Langrage invented the method of variation of 
parameters.
› Consider differential equation of the form f(D)y=X.
› when X is of the form ?? ????
,sin???? ,cos???? ,?? ?? ,?? ????
.?? or any 
function of x, then the shortcut methods are available 
which will discuss later on. If X be of any other form say 
tan?? ,sec?? ,csc?? ?????? .,then we have to use one of the 
following methods.
I. The method of partial fractions
II. The method of variation of parameters
Page 2


Method of Variation of Parameters
› Langrage invented the method of variation of 
parameters.
› Consider differential equation of the form f(D)y=X.
› when X is of the form ?? ????
,sin???? ,cos???? ,?? ?? ,?? ????
.?? or any 
function of x, then the shortcut methods are available 
which will discuss later on. If X be of any other form say 
tan?? ,sec?? ,csc?? ?????? .,then we have to use one of the 
following methods.
I. The method of partial fractions
II. The method of variation of parameters
Continue…
› Consider the second order linear differential equation 
with constant co-efficient
?? 2
?? ?? ?? 2
+?? 1
????
????
+?? 2
?? =?? ……(i)
› Let the complementary function of (i) be 
?? =?? 1
?? 1
+?? 2
?? 2
› Then ?? 1
and ?? 2
satisfy the equation
?? 2
?? ?? ?? 2
+?? 1
????
????
+?? 2
?? =0 ……(ii)
Page 3


Method of Variation of Parameters
› Langrage invented the method of variation of 
parameters.
› Consider differential equation of the form f(D)y=X.
› when X is of the form ?? ????
,sin???? ,cos???? ,?? ?? ,?? ????
.?? or any 
function of x, then the shortcut methods are available 
which will discuss later on. If X be of any other form say 
tan?? ,sec?? ,csc?? ?????? .,then we have to use one of the 
following methods.
I. The method of partial fractions
II. The method of variation of parameters
Continue…
› Consider the second order linear differential equation 
with constant co-efficient
?? 2
?? ?? ?? 2
+?? 1
????
????
+?? 2
?? =?? ……(i)
› Let the complementary function of (i) be 
?? =?? 1
?? 1
+?? 2
?? 2
› Then ?? 1
and ?? 2
satisfy the equation
?? 2
?? ?? ?? 2
+?? 1
????
????
+?? 2
?? =0 ……(ii)
Continue…
› Let us assume that the particular integral of (i) be
?? =?? ?? 1
+?? ?? 2
…… (iii)
where u and v are unknown functions of x
› Differentiating w.r. to x, we have
?? '
=?? ?? 1
'
+?? '
?? 1
+?? ?? 2
'
+?? '
?? 2
…… (iv)
› To determine two unknown functions u and v, we need two 
equations.
We assume that ?? '
?? 1
+?? '
?? 2
=0 …… (v)
?(iv) reduces to
?? '
=?? ?? 1
'
+?? ?? 2
'
…… (vi)
Page 4


Method of Variation of Parameters
› Langrage invented the method of variation of 
parameters.
› Consider differential equation of the form f(D)y=X.
› when X is of the form ?? ????
,sin???? ,cos???? ,?? ?? ,?? ????
.?? or any 
function of x, then the shortcut methods are available 
which will discuss later on. If X be of any other form say 
tan?? ,sec?? ,csc?? ?????? .,then we have to use one of the 
following methods.
I. The method of partial fractions
II. The method of variation of parameters
Continue…
› Consider the second order linear differential equation 
with constant co-efficient
?? 2
?? ?? ?? 2
+?? 1
????
????
+?? 2
?? =?? ……(i)
› Let the complementary function of (i) be 
?? =?? 1
?? 1
+?? 2
?? 2
› Then ?? 1
and ?? 2
satisfy the equation
?? 2
?? ?? ?? 2
+?? 1
????
????
+?? 2
?? =0 ……(ii)
Continue…
› Let us assume that the particular integral of (i) be
?? =?? ?? 1
+?? ?? 2
…… (iii)
where u and v are unknown functions of x
› Differentiating w.r. to x, we have
?? '
=?? ?? 1
'
+?? '
?? 1
+?? ?? 2
'
+?? '
?? 2
…… (iv)
› To determine two unknown functions u and v, we need two 
equations.
We assume that ?? '
?? 1
+?? '
?? 2
=0 …… (v)
?(iv) reduces to
?? '
=?? ?? 1
'
+?? ?? 2
'
…… (vi)
Continue…
› Differentiating w.r. to x, we get
?? "
=?? ?? 1
"
+?? '
?? 1
'
+?? ?? 2
"
+?? '
?? 2
'
› Substituting the values of ?? ,?? '
and ?? "
in (i), we have
?? ?? 1
"
+?? 1
?? 1
'
+?? 2
?? 1
+?? ?? 2
"
+?? 1
?? 2
'
+?? 2
?? 2
+?? '
?? 1
'
+?? '
?? 2
'
=?? ……(vii)
› But ?? 1
and ?? 2
satisfy equation (ii)
??? 1
"
+?? 1
?? 1
'
+?? 2
?? 1
=0and ?? 2
"
+?? 1
?? 2
'
+?? 2
?? 2
=0
› Equation (viii) takes the form,
?? '
?? 1
'
+?? '
?? 2
'
=?? ……(viii)
Page 5


Method of Variation of Parameters
› Langrage invented the method of variation of 
parameters.
› Consider differential equation of the form f(D)y=X.
› when X is of the form ?? ????
,sin???? ,cos???? ,?? ?? ,?? ????
.?? or any 
function of x, then the shortcut methods are available 
which will discuss later on. If X be of any other form say 
tan?? ,sec?? ,csc?? ?????? .,then we have to use one of the 
following methods.
I. The method of partial fractions
II. The method of variation of parameters
Continue…
› Consider the second order linear differential equation 
with constant co-efficient
?? 2
?? ?? ?? 2
+?? 1
????
????
+?? 2
?? =?? ……(i)
› Let the complementary function of (i) be 
?? =?? 1
?? 1
+?? 2
?? 2
› Then ?? 1
and ?? 2
satisfy the equation
?? 2
?? ?? ?? 2
+?? 1
????
????
+?? 2
?? =0 ……(ii)
Continue…
› Let us assume that the particular integral of (i) be
?? =?? ?? 1
+?? ?? 2
…… (iii)
where u and v are unknown functions of x
› Differentiating w.r. to x, we have
?? '
=?? ?? 1
'
+?? '
?? 1
+?? ?? 2
'
+?? '
?? 2
…… (iv)
› To determine two unknown functions u and v, we need two 
equations.
We assume that ?? '
?? 1
+?? '
?? 2
=0 …… (v)
?(iv) reduces to
?? '
=?? ?? 1
'
+?? ?? 2
'
…… (vi)
Continue…
› Differentiating w.r. to x, we get
?? "
=?? ?? 1
"
+?? '
?? 1
'
+?? ?? 2
"
+?? '
?? 2
'
› Substituting the values of ?? ,?? '
and ?? "
in (i), we have
?? ?? 1
"
+?? 1
?? 1
'
+?? 2
?? 1
+?? ?? 2
"
+?? 1
?? 2
'
+?? 2
?? 2
+?? '
?? 1
'
+?? '
?? 2
'
=?? ……(vii)
› But ?? 1
and ?? 2
satisfy equation (ii)
??? 1
"
+?? 1
?? 1
'
+?? 2
?? 1
=0and ?? 2
"
+?? 1
?? 2
'
+?? 2
?? 2
=0
› Equation (viii) takes the form,
?? '
?? 1
'
+?? '
?? 2
'
=?? ……(viii)
Continue…
› Solving (v) and (ix), we get
?? '
=
?? 2
?? ?? 1
?? 2
'
-?? 2
?? 1
'
and ?? '
=
?? 1
?? ?? 1
?? 2
'
-?? 2
?? 1
'
› Integrating, we get 
?? =-
 
?? 2
?? ?? ???? and v=-
 
?? 1
?? ?? ???? , where ?? =?? 1
?? 2
'
-?? 2
?? 1
'
› Substituting the values of u and v in (iii), we have
?? .?? =?? =-?? 1
 
?? 2
?? ?? ???? +?? 2
 
?? 1
?? ?? ???? ……(ix)