introduction to differential equations...ordinary differential equations partial differential...
TRANSCRIPT
Introduction to Differential
Equations
Mr. Raut S. R. Mr. K.S.K.College Beed
Definition:
A differential equation is an equation containing an unknown function
and its derivatives.
32 xdx
dy
032
2
aydx
dy
dx
yd
36
4
3
3
y
dx
dy
dx
yd
Examples:.
y is dependent variable and x is independent variable,
and these are ordinary differential equations
1.
2.
3.
ordinary differential equations
Partial Differential Equation
Examples:
02
2
2
2
y
u
x
u
04
4
4
4
t
u
x
u
t
u
t
u
x
u
2
2
2
2
u is dependent variable and x and y are independent variables,
and is partial differential equation.
u is dependent variable and x and t are independent variables
1.
2.
3.
Order of Differential Equation
The order of the differential equation is order of the highest
derivative in the differential equation.
Differential Equation ORDER
32 xdx
dy
0932
2
ydx
dy
dx
yd
36
4
3
3
y
dx
dy
dx
yd
1
2
3
Degree of Differential Equation
Differential Equation Degree
03
2
2
aydx
dy
dx
yd
36
4
3
3
y
dx
dy
dx
yd
03
53
2
2
dx
dy
dx
yd
1
1
3
The degree of a differential equation is power of the highest
order derivative term in the differential equation.
Linear Differential Equation
A differential equation is linear, if
1. dependent variable and its derivatives are of degree one,
2. coefficients of a term does not depend upon dependent
variable.
Example:
36
4
3
3
y
dx
dy
dx
yd
is non - linear because in 2nd term is not of degree one.
.0932
2
ydx
dy
dx
ydExample:
is linear.
1.
2.
Example: 3
2
22 x
dx
dyy
dx
ydx
is non - linear because in 2nd term coefficient depends on y.
3.
Example:
is non - linear because
ydx
dysin
!3
sin3y
yy is non – linear
4.
It is Ordinary/partial Differential equation of order… and of degree…, it is
linear / non linear, with independent variable…, and dependent variable….
1st – order differential equation
2. Differential form:
01 ydxdyx.
.0),,( dx
dyyxf),( yxf
dx
dy
3. General form:
or
1. Derivative form:
xgyxadx
dyxa 01
9 Differential Equation
First Order Ordinary Differential
equation
10 Differential Equation
Second order Ordinary Differential
Equation
11 Differential Equation
nth – order linear differential
equation 1. nth – order linear differential equation with constant coefficients.
xgyadx
dya
dx
yda
dx
yda
dx
yda
n
n
nn
n
n
012
2
21
1
1 ....
2. nth – order linear differential equation with variable coefficients
xgyxadx
dyxa
dx
ydxa
dx
ydxa
dx
dyxa
n
n
nn
012
2
2
1
1 ......
12 Differential Equation
Differential Equation 13
Solution of Differential Equation
y=3x+c1 , is solution of the 1st order differential equation , c1 is arbitrary constant.
As is solution of the differential equation for every value of c1, hence it is known as general solution.
3dx
dy
Examples
Examples
sin cosy x y x C
2 3
1 1 26 e 3 e ex x xy x y x C y x C x C
Observe that the set of solutions to the above 1st order equation has 1 parameter,
while the solutions to the above 2nd order equation depend on two parameters.
Diffe
ren
tia
l E
qu
atio
n
Families of Solutions 9 ' 4 0yy x
1 19 ' 4 9 ( ) '( ) 4yy x dx C y x y x dx xdx C
2 2
1This yields where .4 9 18
Cy xC C
Example
Solution
The solution is a family of ellipses.
2
2 2 2 2
1 1 1
99 2 2 9 4 2
2
yydy x C x C y x C
Observe that given any point (x0,y0),
there is a unique solution curve of the
above equation which curve goes
through the given point.
Diffe
ren
tia
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qu
atio
n
Origin of Differential Equations
Solution
1.Geometric Origin
21 cxcy
02
2
dx
yd
1. For the family of straight lines
the differential equation is
. 2. For the family of curves
2
2x
cey the differential equation is xy
dx
dy
A.
B. xx ececy 3
2
2
1
the differential equation is 06
2
2
ydx
dy
dx
yd
16
Diffe
ren
tia
l E
qu
atio
n
Physical Origin
1. Free falling stone g
dt
sd
2
2
where s is distance or height and
g is acceleration due to gravity.
2. Spring vertical displacement ky
dt
ydm
2
2
where y is displacement,
m is mass and
k is spring constant
3. RLC – circuit, Kirchoff ’s Second Law
Eqcdt
dqR
dt
qdL
12
2
q is charge on
capacitor,
L is inductance,
c is capacitance.
R is resistance and
E is voltage 17
Diffe
ren
tia
l E
qu
atio
n
Physical Origin
1.Newton’s Low of Cooling sTTdt
dT
where dt
dT is rate of cooling of the liquid,
sTT is temperature difference between the liquid ‘T’
and its surrounding Ts
dy
ydt
2. Growth and Decay
y is the quantity present at any time
18 Differential Equation
Diffe
ren
tia
l E
qu
atio
n
Thanking you
Differential Equation 19