UPSC Exam > UPSC Notes > Mathematics Optional Notes for UPSC > Laplace and inverse Laplace Transformation and Properties
Laplace and inverse Laplace Transformation and Properties | Mathematics Optional Notes for UPSC PDF Download
| Download, print and study this document offline |
Please wait while the PDF view is loading
Page 1
Edurev123
8. Laplace and inverse Laplace
Transformation and Properties
8.1 Find the inverse Laplace transform of ?? (?? )=???? (
?? +?? ?? +?? ) .
(2009: 20 Marks)
Solution:
Approach: Use the differentiation property of inverse Laplace transform.
Given:
?? (?? )=ln (
?? +1
?? +?? )
By differentiation property we have
?? -1
[?? (?? )] =
1
?? ?? -1
[?? -1
(?? )]
? ?? -1
[ln (
?? +1
?? +?? )] =
1
?? ?? -1
[
?? ????
ln (
?? +1
?? +?? )]
=
1
?? ?? -1
[
?? ????
ln (?? +1)-ln (?? +?? )]
=
1
?? ?? -1
[
1
?? +1
-
1
?? +?? ]
=
1
?? {?? -1
(
1
?? +1
)-?? -1
(
1
?? +?? )} (Linearity)
=
1
?? {?? -?? -?? -5?? }.
8.2 Use Laplace transform method to solve the following initial value problem:
?? ?? ?? ?? ?? ?? -?? ????
????
+?? =?? ?? ,?? (?? )=?? and
????
????
|
?? =?? =-??
(2011: 15 Marks)
Solution:
The given initial value problem is
Page 2
Edurev123
8. Laplace and inverse Laplace
Transformation and Properties
8.1 Find the inverse Laplace transform of ?? (?? )=???? (
?? +?? ?? +?? ) .
(2009: 20 Marks)
Solution:
Approach: Use the differentiation property of inverse Laplace transform.
Given:
?? (?? )=ln (
?? +1
?? +?? )
By differentiation property we have
?? -1
[?? (?? )] =
1
?? ?? -1
[?? -1
(?? )]
? ?? -1
[ln (
?? +1
?? +?? )] =
1
?? ?? -1
[
?? ????
ln (
?? +1
?? +?? )]
=
1
?? ?? -1
[
?? ????
ln (?? +1)-ln (?? +?? )]
=
1
?? ?? -1
[
1
?? +1
-
1
?? +?? ]
=
1
?? {?? -1
(
1
?? +1
)-?? -1
(
1
?? +?? )} (Linearity)
=
1
?? {?? -?? -?? -5?? }.
8.2 Use Laplace transform method to solve the following initial value problem:
?? ?? ?? ?? ?? ?? -?? ????
????
+?? =?? ?? ,?? (?? )=?? and
????
????
|
?? =?? =-??
(2011: 15 Marks)
Solution:
The given initial value problem is
?? 2
?? ?? ?? 2
-2
????
????
+?? =?? ?? ,?? (0)=2 (??)
(
????
????
)
?? =0
=-1
Taking Laplace transform of both sides of (i), we get
?? (
?? 2
?? ?? ?? 2
-2
????
????
+?? )=?? [?? '
]
? ?? {?? ''
(?? )}-2?? {?? '
(?? )}+?? {?? (?? )}=?? {?? ?? }
? ?? 2
?? {?? (?? )}-???? (0)-?? '
(0)-2[???? {?? (?? )}-?? (0)]+?? {?? (?? )}=
1
??
? ?? (?? )][?? 2
-2?? +1]-?? (2)-(-1)+2(2)=
1
??
? ?? {?? (?? )}(?? -1)
2
=
1
?? +2?? -5
=
1+2?? 2
-5?? ??
? ?? {?? (?? )} =
1+2?? 2
-5?? ?? (?? -1)
2
=
1
?? -1
-
2
(?? -1)
2
+
1
?? … (ii) (By using Partial Fractions)
Taking inverse Laplace of (ii), we get
?? (?? )=?? -1
{
1
?? -1
-
2
(?? -1)
2
+
1
?? }
? ?? (?? )=?? ?? -2?? ?? ?? +1
8.3 Using Laplace transforms, solve the initial value problem
?? ''
+?? ?? +?? =?? -?? ,?? (?? )=-?? ,?? (?? )=??
(2012: 12 Marks)
Solution:
Given :
?? '
+2?? +?? =?? -?? (??)
?? (0) =-1,?? '
(0)=1
Page 3
Edurev123
8. Laplace and inverse Laplace
Transformation and Properties
8.1 Find the inverse Laplace transform of ?? (?? )=???? (
?? +?? ?? +?? ) .
(2009: 20 Marks)
Solution:
Approach: Use the differentiation property of inverse Laplace transform.
Given:
?? (?? )=ln (
?? +1
?? +?? )
By differentiation property we have
?? -1
[?? (?? )] =
1
?? ?? -1
[?? -1
(?? )]
? ?? -1
[ln (
?? +1
?? +?? )] =
1
?? ?? -1
[
?? ????
ln (
?? +1
?? +?? )]
=
1
?? ?? -1
[
?? ????
ln (?? +1)-ln (?? +?? )]
=
1
?? ?? -1
[
1
?? +1
-
1
?? +?? ]
=
1
?? {?? -1
(
1
?? +1
)-?? -1
(
1
?? +?? )} (Linearity)
=
1
?? {?? -?? -?? -5?? }.
8.2 Use Laplace transform method to solve the following initial value problem:
?? ?? ?? ?? ?? ?? -?? ????
????
+?? =?? ?? ,?? (?? )=?? and
????
????
|
?? =?? =-??
(2011: 15 Marks)
Solution:
The given initial value problem is
?? 2
?? ?? ?? 2
-2
????
????
+?? =?? ?? ,?? (0)=2 (??)
(
????
????
)
?? =0
=-1
Taking Laplace transform of both sides of (i), we get
?? (
?? 2
?? ?? ?? 2
-2
????
????
+?? )=?? [?? '
]
? ?? {?? ''
(?? )}-2?? {?? '
(?? )}+?? {?? (?? )}=?? {?? ?? }
? ?? 2
?? {?? (?? )}-???? (0)-?? '
(0)-2[???? {?? (?? )}-?? (0)]+?? {?? (?? )}=
1
??
? ?? (?? )][?? 2
-2?? +1]-?? (2)-(-1)+2(2)=
1
??
? ?? {?? (?? )}(?? -1)
2
=
1
?? +2?? -5
=
1+2?? 2
-5?? ??
? ?? {?? (?? )} =
1+2?? 2
-5?? ?? (?? -1)
2
=
1
?? -1
-
2
(?? -1)
2
+
1
?? … (ii) (By using Partial Fractions)
Taking inverse Laplace of (ii), we get
?? (?? )=?? -1
{
1
?? -1
-
2
(?? -1)
2
+
1
?? }
? ?? (?? )=?? ?? -2?? ?? ?? +1
8.3 Using Laplace transforms, solve the initial value problem
?? ''
+?? ?? +?? =?? -?? ,?? (?? )=-?? ,?? (?? )=??
(2012: 12 Marks)
Solution:
Given :
?? '
+2?? +?? =?? -?? (??)
?? (0) =-1,?? '
(0)=1
Taking Laplace transform of both sides of (i),
?? (?? '
(?? ))+2?? (?? (?? ))+?? (?? )=?? (?? -?? )
??? 2
?? (?? )-???? (0)-?? (0)+2(???? (?? )-?? (0))+?? (?? )=
1
?? +1
??? (?? )(?? 2
+2?? +1)-?? (-1)-1-2(-1)=
1
?? +1
? ?? (?? )(?? +1)
2
+(?? +1)=
1
?? +1
? ?? (?? )=
1
(?? +1)
3
-
1
(?? +1)
(???? )
Taking inverse Laplace transform on both sides of (ii), we have
?? =?? -1
[
1
(?? +1)
3
-
1
(?? +1)
]
? ?? =?? -?? ·
?? 2
2
-?? -??
8.4 By using Laplace transform method solve the differential equation :
(?? ?? +?? ?? )?? =?? ?????? (???? +?? ) , ?? ?? =
?? ?? ?? ?? ?? subject to the initial conditions ?? =?? and
????
????
=??
at ?? =?? in which ?? ,?? and ?? are constants.
(2013 : 15 Marks)
Solution:
(?? 2
+?? 2
)?? =?? sin (???? +?? )
Taking Laplace transform on both sides
?? (
?? 2
?? ?? ?? 2
)+?? 2
?? (?? )=???? [sin ???? cos ?? +cos ???? sin ?? ]
? ?? 2
?? (?? )-???? (0)-?? '
(0)+?? 2
?? (?? )=?? [
?? ?? 2
+?? 2
cos ?? +
?? ?? 2
+?? 2
sin ?? ]
? ?? (?? )=?? [
?? (?? 2
+?? 2
)
2
cos ?? +
?? (?? 2
+?? 2
)
2
sin ?? ]
? ?? =?? {?? -1
[
?? (?? 2
+?? 2
)
2
]cos ?? +?? -1
[
?? (?? 2
+?? 2
)
2
]sin ?? }
Page 4
Edurev123
8. Laplace and inverse Laplace
Transformation and Properties
8.1 Find the inverse Laplace transform of ?? (?? )=???? (
?? +?? ?? +?? ) .
(2009: 20 Marks)
Solution:
Approach: Use the differentiation property of inverse Laplace transform.
Given:
?? (?? )=ln (
?? +1
?? +?? )
By differentiation property we have
?? -1
[?? (?? )] =
1
?? ?? -1
[?? -1
(?? )]
? ?? -1
[ln (
?? +1
?? +?? )] =
1
?? ?? -1
[
?? ????
ln (
?? +1
?? +?? )]
=
1
?? ?? -1
[
?? ????
ln (?? +1)-ln (?? +?? )]
=
1
?? ?? -1
[
1
?? +1
-
1
?? +?? ]
=
1
?? {?? -1
(
1
?? +1
)-?? -1
(
1
?? +?? )} (Linearity)
=
1
?? {?? -?? -?? -5?? }.
8.2 Use Laplace transform method to solve the following initial value problem:
?? ?? ?? ?? ?? ?? -?? ????
????
+?? =?? ?? ,?? (?? )=?? and
????
????
|
?? =?? =-??
(2011: 15 Marks)
Solution:
The given initial value problem is
?? 2
?? ?? ?? 2
-2
????
????
+?? =?? ?? ,?? (0)=2 (??)
(
????
????
)
?? =0
=-1
Taking Laplace transform of both sides of (i), we get
?? (
?? 2
?? ?? ?? 2
-2
????
????
+?? )=?? [?? '
]
? ?? {?? ''
(?? )}-2?? {?? '
(?? )}+?? {?? (?? )}=?? {?? ?? }
? ?? 2
?? {?? (?? )}-???? (0)-?? '
(0)-2[???? {?? (?? )}-?? (0)]+?? {?? (?? )}=
1
??
? ?? (?? )][?? 2
-2?? +1]-?? (2)-(-1)+2(2)=
1
??
? ?? {?? (?? )}(?? -1)
2
=
1
?? +2?? -5
=
1+2?? 2
-5?? ??
? ?? {?? (?? )} =
1+2?? 2
-5?? ?? (?? -1)
2
=
1
?? -1
-
2
(?? -1)
2
+
1
?? … (ii) (By using Partial Fractions)
Taking inverse Laplace of (ii), we get
?? (?? )=?? -1
{
1
?? -1
-
2
(?? -1)
2
+
1
?? }
? ?? (?? )=?? ?? -2?? ?? ?? +1
8.3 Using Laplace transforms, solve the initial value problem
?? ''
+?? ?? +?? =?? -?? ,?? (?? )=-?? ,?? (?? )=??
(2012: 12 Marks)
Solution:
Given :
?? '
+2?? +?? =?? -?? (??)
?? (0) =-1,?? '
(0)=1
Taking Laplace transform of both sides of (i),
?? (?? '
(?? ))+2?? (?? (?? ))+?? (?? )=?? (?? -?? )
??? 2
?? (?? )-???? (0)-?? (0)+2(???? (?? )-?? (0))+?? (?? )=
1
?? +1
??? (?? )(?? 2
+2?? +1)-?? (-1)-1-2(-1)=
1
?? +1
? ?? (?? )(?? +1)
2
+(?? +1)=
1
?? +1
? ?? (?? )=
1
(?? +1)
3
-
1
(?? +1)
(???? )
Taking inverse Laplace transform on both sides of (ii), we have
?? =?? -1
[
1
(?? +1)
3
-
1
(?? +1)
]
? ?? =?? -?? ·
?? 2
2
-?? -??
8.4 By using Laplace transform method solve the differential equation :
(?? ?? +?? ?? )?? =?? ?????? (???? +?? ) , ?? ?? =
?? ?? ?? ?? ?? subject to the initial conditions ?? =?? and
????
????
=??
at ?? =?? in which ?? ,?? and ?? are constants.
(2013 : 15 Marks)
Solution:
(?? 2
+?? 2
)?? =?? sin (???? +?? )
Taking Laplace transform on both sides
?? (
?? 2
?? ?? ?? 2
)+?? 2
?? (?? )=???? [sin ???? cos ?? +cos ???? sin ?? ]
? ?? 2
?? (?? )-???? (0)-?? '
(0)+?? 2
?? (?? )=?? [
?? ?? 2
+?? 2
cos ?? +
?? ?? 2
+?? 2
sin ?? ]
? ?? (?? )=?? [
?? (?? 2
+?? 2
)
2
cos ?? +
?? (?? 2
+?? 2
)
2
sin ?? ]
? ?? =?? {?? -1
[
?? (?? 2
+?? 2
)
2
]cos ?? +?? -1
[
?? (?? 2
+?? 2
)
2
]sin ?? }
Now
?? ????
1
(?? 2
+?? 2
)
=-
2?? (?? 2
+?? 2
)
2
? ?? -1
(
2?? (?? 2
+?? 2
)
2
)=(-1)?? ?? -1
(
1
?? 2
+?? 2
)
=
-?? ?? ·sin ????
? ?? -1
(
?? (?? 2
+?? 2
)
2
)=
?? 2?? sin ????
Let
?? (?? ) =
1
?? 2
+?? 2
;?? (?? )=
1
?? 2
+?? 2
?? (?? ) =?? -1
(?? (?? ))=
1
?? sin ????
?? (?? ) =?? -1
(?? (?? ))=
1
?? sin ????
?? -1
(?? (?? )?? (?? )) =?? *?? =??
?? 0
??? (?? )?? (?? -?? )????
=
1
?? 2
??
?? 0
?sin ???? sin ?? (?? -?? )????
=
1
2?? 2
??
?? 0
?(cos ?? (?? -2?? )-cos ???? )????
=
1
2?? 2
[
sin ?? (?? -2?? )
-2?? -?? cos ???? ]
0
?? =
1
2?? 2
[
-sin ????
-2?? +
sin ????
2?? -?? cos ???? ]=
1
2?? 2
[
sin ????
?? -?? cos ???? ]
?? =?? {
1
2?? [
sin ????
?? -?? cos ???? ]cos ?? +(
?? 2?? sin ???? sin ?? )}
=?? {
1
2?? 2
sin ???? cos ?? -
?? 2?? cos (???? +?? )}
=
?? 2?? 2
{sin ???? cos ?? -???? cos (???? +?? )}
8.5 Solve the initial value problem
?? ?? ?? ?? ?? ?? +?? =?? ?? -?? ?? ?????? ?? ,?? (?? )=?? ,?? (?? )=??
by using Laplace-transform.
(2014 : 20 Marks)
Solution:
Given equation is
Page 5
Edurev123
8. Laplace and inverse Laplace
Transformation and Properties
8.1 Find the inverse Laplace transform of ?? (?? )=???? (
?? +?? ?? +?? ) .
(2009: 20 Marks)
Solution:
Approach: Use the differentiation property of inverse Laplace transform.
Given:
?? (?? )=ln (
?? +1
?? +?? )
By differentiation property we have
?? -1
[?? (?? )] =
1
?? ?? -1
[?? -1
(?? )]
? ?? -1
[ln (
?? +1
?? +?? )] =
1
?? ?? -1
[
?? ????
ln (
?? +1
?? +?? )]
=
1
?? ?? -1
[
?? ????
ln (?? +1)-ln (?? +?? )]
=
1
?? ?? -1
[
1
?? +1
-
1
?? +?? ]
=
1
?? {?? -1
(
1
?? +1
)-?? -1
(
1
?? +?? )} (Linearity)
=
1
?? {?? -?? -?? -5?? }.
8.2 Use Laplace transform method to solve the following initial value problem:
?? ?? ?? ?? ?? ?? -?? ????
????
+?? =?? ?? ,?? (?? )=?? and
????
????
|
?? =?? =-??
(2011: 15 Marks)
Solution:
The given initial value problem is
?? 2
?? ?? ?? 2
-2
????
????
+?? =?? ?? ,?? (0)=2 (??)
(
????
????
)
?? =0
=-1
Taking Laplace transform of both sides of (i), we get
?? (
?? 2
?? ?? ?? 2
-2
????
????
+?? )=?? [?? '
]
? ?? {?? ''
(?? )}-2?? {?? '
(?? )}+?? {?? (?? )}=?? {?? ?? }
? ?? 2
?? {?? (?? )}-???? (0)-?? '
(0)-2[???? {?? (?? )}-?? (0)]+?? {?? (?? )}=
1
??
? ?? (?? )][?? 2
-2?? +1]-?? (2)-(-1)+2(2)=
1
??
? ?? {?? (?? )}(?? -1)
2
=
1
?? +2?? -5
=
1+2?? 2
-5?? ??
? ?? {?? (?? )} =
1+2?? 2
-5?? ?? (?? -1)
2
=
1
?? -1
-
2
(?? -1)
2
+
1
?? … (ii) (By using Partial Fractions)
Taking inverse Laplace of (ii), we get
?? (?? )=?? -1
{
1
?? -1
-
2
(?? -1)
2
+
1
?? }
? ?? (?? )=?? ?? -2?? ?? ?? +1
8.3 Using Laplace transforms, solve the initial value problem
?? ''
+?? ?? +?? =?? -?? ,?? (?? )=-?? ,?? (?? )=??
(2012: 12 Marks)
Solution:
Given :
?? '
+2?? +?? =?? -?? (??)
?? (0) =-1,?? '
(0)=1
Taking Laplace transform of both sides of (i),
?? (?? '
(?? ))+2?? (?? (?? ))+?? (?? )=?? (?? -?? )
??? 2
?? (?? )-???? (0)-?? (0)+2(???? (?? )-?? (0))+?? (?? )=
1
?? +1
??? (?? )(?? 2
+2?? +1)-?? (-1)-1-2(-1)=
1
?? +1
? ?? (?? )(?? +1)
2
+(?? +1)=
1
?? +1
? ?? (?? )=
1
(?? +1)
3
-
1
(?? +1)
(???? )
Taking inverse Laplace transform on both sides of (ii), we have
?? =?? -1
[
1
(?? +1)
3
-
1
(?? +1)
]
? ?? =?? -?? ·
?? 2
2
-?? -??
8.4 By using Laplace transform method solve the differential equation :
(?? ?? +?? ?? )?? =?? ?????? (???? +?? ) , ?? ?? =
?? ?? ?? ?? ?? subject to the initial conditions ?? =?? and
????
????
=??
at ?? =?? in which ?? ,?? and ?? are constants.
(2013 : 15 Marks)
Solution:
(?? 2
+?? 2
)?? =?? sin (???? +?? )
Taking Laplace transform on both sides
?? (
?? 2
?? ?? ?? 2
)+?? 2
?? (?? )=???? [sin ???? cos ?? +cos ???? sin ?? ]
? ?? 2
?? (?? )-???? (0)-?? '
(0)+?? 2
?? (?? )=?? [
?? ?? 2
+?? 2
cos ?? +
?? ?? 2
+?? 2
sin ?? ]
? ?? (?? )=?? [
?? (?? 2
+?? 2
)
2
cos ?? +
?? (?? 2
+?? 2
)
2
sin ?? ]
? ?? =?? {?? -1
[
?? (?? 2
+?? 2
)
2
]cos ?? +?? -1
[
?? (?? 2
+?? 2
)
2
]sin ?? }
Now
?? ????
1
(?? 2
+?? 2
)
=-
2?? (?? 2
+?? 2
)
2
? ?? -1
(
2?? (?? 2
+?? 2
)
2
)=(-1)?? ?? -1
(
1
?? 2
+?? 2
)
=
-?? ?? ·sin ????
? ?? -1
(
?? (?? 2
+?? 2
)
2
)=
?? 2?? sin ????
Let
?? (?? ) =
1
?? 2
+?? 2
;?? (?? )=
1
?? 2
+?? 2
?? (?? ) =?? -1
(?? (?? ))=
1
?? sin ????
?? (?? ) =?? -1
(?? (?? ))=
1
?? sin ????
?? -1
(?? (?? )?? (?? )) =?? *?? =??
?? 0
??? (?? )?? (?? -?? )????
=
1
?? 2
??
?? 0
?sin ???? sin ?? (?? -?? )????
=
1
2?? 2
??
?? 0
?(cos ?? (?? -2?? )-cos ???? )????
=
1
2?? 2
[
sin ?? (?? -2?? )
-2?? -?? cos ???? ]
0
?? =
1
2?? 2
[
-sin ????
-2?? +
sin ????
2?? -?? cos ???? ]=
1
2?? 2
[
sin ????
?? -?? cos ???? ]
?? =?? {
1
2?? [
sin ????
?? -?? cos ???? ]cos ?? +(
?? 2?? sin ???? sin ?? )}
=?? {
1
2?? 2
sin ???? cos ?? -
?? 2?? cos (???? +?? )}
=
?? 2?? 2
{sin ???? cos ?? -???? cos (???? +?? )}
8.5 Solve the initial value problem
?? ?? ?? ?? ?? ?? +?? =?? ?? -?? ?? ?????? ?? ,?? (?? )=?? ,?? (?? )=??
by using Laplace-transform.
(2014 : 20 Marks)
Solution:
Given equation is
?? 2
?? ?? ?? 2
+?? =8?? -2?? sin ??
? ?? '
+?? =8?? -2?? sin?? (??)
? Taking Laplace transform of both sides of (??) , we get
?? (?? '
)+?? (?? )=8?? (?? -2?? sin ?? )
? ?? 2
?? {(?? )}-???? (0)-?? '
(0)+?? (?? (?? )=
8
(?? +2)
2
+1
? ?? 2
?? {(?? )}+?? {?? (?? )}=
8
?? 2
+4?? +5
? ?? (?? (?? ))(?? 2
+1)=
8
?? 2
+4?? +5
? ?? {?? (?? )}=
8
(?? 2
+1)(?? 2
+4?? +5)
? ?? (?? )=?? -1
8
(?? 2
+1)(?? 2
+4?? +5)
?? (?? )=?? -1
[
-?? +1
?? 2
+1
+
?? +3
?? 2
+4?? +5
]
=?? -1
(
-?? ?? 2
+1
)+?? -1
(
1
?? 2
+1
)+?? -1
(
(?? +2)+1
(?? +2)
2
+1
)
=-cos ?? +sin ?? +?? -2?? ?? -1
(
?? +1
?? 2
+1
)
=-cos ?? +sin ?? +?? -2?? {?? -1
(
?? ?? 2
+1
)··-1(
1
?? 2
+1
)}
=-cos ?? +sin ?? +?? -2?? cos ?? +?? -2?? sin ?? =(?? -2?? -1)cos ?? +(?? -2?? +1)sin ??
which is the required solution.
8.6
(i) Obtain Laplace Inverse transform of
?? (?? )={???? (?? +
?? ?? ?? )+
?? ?? ?? +????
?? -????
}
(2015 : 6 Marks)
(ii) Using Laplace transform, solve:
?? ''
+?? =?? ,?? (?? )=?? ,?? (?? )=-??
(2015 : 6 Marks)
Solution:
Read More
|
387 videos|203 docs
|
Related Exams
About this Document
|
4.88/5 Rating |
|
Dec 22, 2024 Last updated |
Document Description: Laplace and inverse Laplace Transformation and Properties for UPSC 2024 is part of Mathematics Optional Notes for UPSC preparation.
The notes and questions for Laplace and inverse Laplace Transformation and Properties have been prepared according to the UPSC exam syllabus. Information about Laplace and inverse Laplace Transformation and Properties covers topics
like and Laplace and inverse Laplace Transformation and Properties Example, for UPSC 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Laplace and inverse Laplace Transformation and Properties.
Introduction of Laplace and inverse Laplace Transformation and Properties in English is available as part of our Mathematics Optional Notes for UPSC
for UPSC & Laplace and inverse Laplace Transformation and Properties in Hindi for Mathematics Optional Notes for UPSC course.
Download more important topics related with notes, lectures and mock test series for UPSC
Exam by signing up for free. UPSC: Laplace and inverse Laplace Transformation and Properties | Mathematics Optional Notes for UPSC
Description
Full syllabus notes, lecture & questions for Laplace and inverse Laplace Transformation and Properties | Mathematics Optional Notes for UPSC - UPSC | Plus excerises question with solution to help you revise complete syllabus for Mathematics Optional Notes for UPSC | Best notes, free PDF download
Information about Laplace and inverse Laplace Transformation and Properties
In this doc you can find the meaning of Laplace and inverse Laplace Transformation and Properties defined & explained in the simplest way possible. Besides explaining types of
Laplace and inverse Laplace Transformation and Properties theory, EduRev gives you an ample number of questions to practice Laplace and inverse Laplace Transformation and Properties tests, examples and also practice UPSC
tests
|
Explore Courses for UPSC exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
Related Searches
ppt
,practice quizzes
,Previous Year Questions with Solutions
,past year papers
,mock tests for examination
,video lectures
,Laplace and inverse Laplace Transformation and Properties | Mathematics Optional Notes for UPSC
,Laplace and inverse Laplace Transformation and Properties | Mathematics Optional Notes for UPSC
,Semester Notes
,Summary
,study material
,MCQs
,Sample Paper
,Free
,Laplace and inverse Laplace Transformation and Properties | Mathematics Optional Notes for UPSC
,Objective type Questions
,Important questions
,Exam
,Extra Questions
,shortcuts and tricks
,Viva Questions
;
Additional Information about Laplace and inverse Laplace Transformation and Properties for UPSC Preparation
Laplace and inverse Laplace Transformation and Properties Free PDF Download
The Laplace and inverse Laplace Transformation and Properties is an invaluable resource that delves deep into the core of the UPSC exam.
These study notes are curated by experts and cover all the essential topics and concepts, making your preparation more efficient and effective.
With the help of these notes, you can grasp complex subjects quickly, revise important points easily,
and reinforce your understanding of key concepts. The study notes are presented in a concise and easy-to-understand manner,
allowing you to optimize your learning process. Whether you're looking for best-recommended books, sample papers, study material,
or toppers' notes, this PDF has got you covered. Download the Laplace and inverse Laplace Transformation and Properties now and kickstart your journey towards success in the UPSC exam.
Importance of Laplace and inverse Laplace Transformation and Properties
The importance of Laplace and inverse Laplace Transformation and Properties cannot be overstated, especially for UPSC aspirants.
This document holds the key to success in the UPSC exam.
It offers a detailed understanding of the concept, providing invaluable insights into the topic.
By knowing the concepts well in advance, students can plan their preparation effectively.
Utilize this indispensable guide for a well-rounded preparation and achieve your desired results.
Laplace and inverse Laplace Transformation and Properties Notes
Laplace and inverse Laplace Transformation and Properties Notes offer in-depth insights into the specific topic to help you master it with ease.
This comprehensive document covers all aspects related to Laplace and inverse Laplace Transformation and Properties.
It includes detailed information about the exam syllabus, recommended books, and study materials for a well-rounded preparation.
Practice papers and question papers enable you to assess your progress effectively.
Additionally, the paper analysis provides valuable tips for tackling the exam strategically.
Access to Toppers' notes gives you an edge in understanding complex concepts.
Whether you're a beginner or aiming for advanced proficiency, Laplace and inverse Laplace Transformation and Properties Notes on EduRev are your ultimate resource for success.
Laplace and inverse Laplace Transformation and Properties UPSC Questions
The "Laplace and inverse Laplace Transformation and Properties UPSC Questions" guide is a valuable resource for all aspiring students preparing for the
UPSC exam. It focuses on providing a wide range of practice questions to help students gauge
their understanding of the exam topics. These questions cover the entire syllabus, ensuring comprehensive preparation.
The guide includes previous years' question papers for students to familiarize themselves with the exam's format and difficulty level.
Additionally, it offers subject-specific question banks, allowing students to focus on weak areas and improve their performance.
Study Laplace and inverse Laplace Transformation and Properties on the App
Students of UPSC can study Laplace and inverse Laplace Transformation and Properties alongwith tests & analysis from the EduRev app,
which will help them while preparing for their exam. Apart from the Laplace and inverse Laplace Transformation and Properties,
students can also utilize the EduRev App for other study materials such as previous year question papers, syllabus, important questions, etc.
The EduRev App will make your learning easier as you can access it from anywhere you want.
The content of Laplace and inverse Laplace Transformation and Properties is prepared as per the latest UPSC syllabus.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup