DE is an equation containing derivatives. Partial DE : Has more than one independent variable First Order DE, First Degree Second Order DE, First Degree Third Order DE, First Degree Second Order DE, Second Degree

ORDER & DEGREE OF A DE:- :

ORDER & DEGREE OF A DE:- Order of a DE : It is the order of the highest derivative.
Degree of a DE: It is the degree of the highest ordered derivative.
Linear DE: It is a DE in which, the dependant variable and its differential co-efficients occur in it in the first degree only, and are not multiplied together.

Solve the following:- :

Solve the following:- 1. A curve is defined by the condition that at each of its points (x,y), its slope is equal to twice the sum of the co-ordinates of the point. Express the condition by a DE.

Slide 5:

2. 100 gm of sugar in water is converted into dextrose at a rate which is proportional to the amount unconverted. Find the DE expressing the rate of conversion after t minutes.

Slide 6:

3. Obtain the DE associated with the primitive y=Ax2+Bx+C Primitive equation is one which does not have derivatives.
In the eqn given, arbitrary constants are 3 (A,B and C)
To get the required DE we must eliminate the constants.

Examples of DE of Physical systems:- :

Examples of DE of Physical systems:-

SOLUTION OF FIRST ORDER LINEAR DIFFERENTIAL EQUATION (LDE):- :

SOLUTION OF FIRST ORDER LINEAR DIFFERENTIAL EQUATION (LDE):- Finding the relationship between x and y from the DE is known as the solution of a DE.

Case 1: k is positive s=Ae+kt :

Case 1: k is positive s=Ae+kt This will be an increasing exponential & divergent If A is the amplitude at t=0, the time t2 taken for s to double its value is called “time to double amplitude”
It is given by

Case 2: k is Negative s=Ae-kt :

Case 2: k is Negative s=Ae-kt This will be an decreasing exponential & convergent A (½)A s = Ae - kt s t t1/2 If A is the amplitude at t=0, the time t1/2 taken for s to halve its value is called “time to half amplitude”
It is given by

General Solution of a First Order LDE:- :

General Solution of a First Order LDE:-

Slide 14:

First Order LDE represents a family of curves
s=Ae-kt+C
A is known as the parameter of the family.
C is known as the initial bias in the system.
Eg. Initial bias in a strain gauge.

SECOND ORDER LDE:- :

SECOND ORDER LDE:- General form of a 2nd Order LDE with constant co-efficients:-

3 Possible sets of Roots:- :

3 Possible sets of Roots:- Two distinct real roots.
Two identical roots.
Pair of Complex roots.

DISTINCT REAL ROOTS:- :

DISTINCT REAL ROOTS:-

IDENTICAL REAL ROOTS:- :

IDENTICAL REAL ROOTS:-

COMPLEX PAIR OF ROOTS:- :

COMPLEX PAIR OF ROOTS:-

Solve the following:- :

Solve the following:-

General Solution of a First Order LDE:- :

General Solution of a First Order LDE:- Linear DE: It is a DE in which, the dependant variable and its differential co-efficients occur in it in the first degree only, and are not multiplied together. Lebnitz Linear Equation

DIFFERENTIAL EQUATIONS ASSIGNMENT : DOS – 18 Aug 08 :

DIFFERENTIAL EQUATIONS ASSIGNMENT : DOS – 18 Aug 08

DIFFERENTIAL EQUATIONS ASSIGNMENT : DOS – 18 Aug 08 :

DIFFERENTIAL EQUATIONS ASSIGNMENT : DOS – 18 Aug 08

Bernoulli’s Equation:- :

Bernoulli’s Equation:-

Determining values of Arbitrary Constants:- :

Determining values of Arbitrary Constants:- We have seen that the solution of a DE contains one or more arbitrary constants. We will now find these constants; given the initial conditions of the DE.
For a DE with a solution having two arbitrary constants, we require two independent initial conditions to determine the values of the two constants.

Solve the following LDE:- :

Solve the following LDE:-

Significance of a LDE in a Physical System:- :

Significance of a LDE in a Physical System:- Mass (m) Dashpot
Damping Co-eff (b) Spring Const (k) Aircraft Pitching Motion x

Mass – Damper Analogy (2nd Order LDE):- :

Mass – Damper Analogy (2nd Order LDE):-

Case 1 : Zero Damping (b=0) (Inference : Dashpot is removed) :

Case 1 : Zero Damping (b=0) (Inference : Dashpot is removed)

Problem:- :

Problem:- A Sopwith Camel moved 0.5 m during its routine tyre check. On braking its oscillations were of the order of 6 sec. If the mass of the aircraft was to be assumed as 300 kg,
(a) Determine the stiffness of the oleo spring.
(b) Plot the response of the aircraft. Note: If nothing is given wrt oscillations, consider the case as one with free oscillations without damping.

Case 2 : Low Damping (b≠0 & a small value) :

Case 2 : Low Damping (b≠0 & a small value)

Case 3 : High Damping (b≠0 & a high value) :

Case 3 : High Damping (b≠0 & a high value)

Case 4 : Critical Damping :

Case 4 : Critical Damping Critical Damping is defined as the condition wherein the damping is just sufficient to prevent oscillatory response.
The necessary condition is that (b2-4mk)=0
Critical Damping co-efficient

Critical Damping Response:- :

Critical Damping Response:- x0 x t

Relative Damping Co-efficient:- :

Relative Damping Co-efficient:-

2nd order LDE Response wrt & n:- :

2nd order LDE Response wrt & n:-

Zero Stiffness (k=0):- :

Zero Stiffness (k=0):-

ROOT LOCUS PLOTS/ ARGAND DIAGRAM:- (STABILITY DOMAIN) :

ROOT LOCUS PLOTS/ ARGAND DIAGRAM:- (STABILITY DOMAIN) Study of stability characteristics with variabtion of b, m & k is undertaken by plotting the values of the roots of the 2nd order LDE =j

ROOT LOCUS PLOTS/ ARGAND DIAGRAM:- :

ROOT LOCUS PLOTS/ ARGAND DIAGRAM:-

You do not have the permission to view this presentation. In order to view it, please
contact the author of the presentation.

Send to Blogs and Networks

Processing ....

Premium member

Use HTTPs

HTTPS (Hypertext Transfer Protocol Secure) is a protocol used by Web servers to transfer and display Web content securely. Most web browsers block content or generate a “mixed content” warning when users access web pages via HTTPS that contain embedded content loaded via HTTP. To prevent users from facing this, Use HTTPS option.

By: iltafhussain1800 (16 month(s) ago)

Please give me download link