2.1 Introduction to DE
2.2 Concept of Solution
2.3Separation of Variable
2.4 Homogeneous Eq
2.5 Linear Eq
2.6 Exact Eq
2.7 Application of 1st
2.1 INTRODUCTION
2.1 INTRODUCTION
 1st order differential equation :-
 2nd order differential equation :-
 nth order differential equation :-
2.2 CONCEPT OF SOLUTION
• If the solution of nth order ODE contains n arbitrary constants, then the
solution is called GENERAL SOLUTION of the differential equation.
• A solution of nth order ODE without any arbitrary constant is called
PARTICULAR SOLUTION.
•The functional relationship between the independent variable and the
dependent variable (such as y = f(x)) which satisfies the given differential
equation is called the solution of the differential equation.
2.2 CONCEPT OF SOLUTION
Exercises
1) Given that y  25 y  0 . Show that
a) y  cos 5 x and y  sin 5 x is a solution.
b) y  C cos 5 x  D sin 5 x , C and D are constants is also a solution.
2
2) If y  Ax 
B
2
, prove that x y  2 y .
x
1ST ORDER OF DE
 There are 4 types of 1st order differential equations:-
i) Separable equations
ii) Homogeneous equations
iii) Linear equations
iv) Exact equations
2.3 SEPARABLE DE
 Write the eqn in the form of
 Integrate both sides
 Simplify the solution
2.3SEPARABLE DE
 Example
Solve the following differential equation:
dy 4 x
 3
dx 3 y
Find the particular solution of the following DE when
y (0)  2 :
1 dy
2

xy
e x dx
2.3SEPARABLE DE
 Example
Solve the following differential equation:
dy
 (1  e  x )( y 2  1)
dx
Find the particular solution of the following DE :
y
dy
 tan 2 x ; y (0)  2
dx
2.3SEPARABLE DE
2.4 HOMOGENEOUS EQUATIONS
 • General form:
dy
 F ( x, y )
dx
 •Method of solution:
1. Show that F ( x,  y )  F ( x, y )
dy
dv
2. Substitute y  xv and
 x  v into the general form of
dx
dx
homogeneous equation.
3. Separate variables x and v to form separable equation.
4. Solve the separable equation.
y
5. Re-Substitute v  into solution in step 4 and simplify the
x
solution.
2.4 HOMOGENEOUS EQUATIONS
Example a: Show that the DE is the homogeneous equation
dy 2 xy  3 y 2
 2
dx
x  2 xy
Example b: Solve the DE
dy x 2  y 2

dx
2 x2
Example c: Solve the DE
( x  y )dx  xdy  0
2.4 HOMOGENEOUS EQUATIONS
Example d:
Solve the differential equation
dy x3  y 3

dx
xy 2
with condition y = 2 when x = 2.
Example e: . By using x  X and y  Y  2 , solve
.
dy x  y  2

dx x  y  2
2.4 HOMOGENEOUS EQUATIONS
Exercises
Verify that each of the following equations is homogeneous and then solve it.
a)
( y  2 x  y dx  xdy  0
2
2
Hint :

1
1  x2
dx  sinh 1 x
b) ( xy  y 2 )dx  x( x  3 y )dy  0
y
x
c)
xydy  ( x 2 e  y 2 )dx  0
d)
dy 4 y 2  x 2

; y(1)  1
dx
2 xy
e) ( x 3  y 2
f)
x 2  y 2 )dx  xy x 2  y 2 dy  0 ; y (1)  1
y 4 dx  ( x 4  xy3 )dy  0 ; y(1)  2
2.5 LINEAR EQUATIONS
dy
 General form:- a( x)  b( x) y  c( x)
dx
 To solve:-
b( x )
dy
p
(
x
)

arrange the eqn. to form  p( x) y  q( x) where
a( x)
dx
c( x)
and q( x) 
a ( x)
p ( x ) dx
1st,
2nd, obtain the integrating factor:   e 
3rd, multiply the integrating factor with 1st eqn with  to become
dy
  p( x) y   q ( x) 
dx
4th ,The eqn can be written in the form of
d
 y   q( x) 
dx
5th , Simplify to y    q ( x) dx
Finally, the general solution of linear equation is
1
y
q( x) dx


2.5 LINEAR EQUATIONS
Example a: Solve the equation
Example b: Solve the equation
Example c: Solve the equation
Example d: Solve the equation
Example e: Solve
(1  x 2 )
dy
 y  x3
dx
dy
 y  2e x
dx
dy
xy
 2
dx
x 9
x
dy
 y tan x  cos x; y (0)  1
dx
dy
 xy  1
dx
2.5 LINEAR EQUATIONS
Exercises
Find the general solution of the following linear equations.
a) dy  y  2 x  1
dx x
dy
1
y
dx
x( x 2  1)
b)
x
c)
dy
2x
2

y

x
dx 1  x 2
d)
e)
f)
dy
 3
 y cot x  cot x ; y 
dx
 2
dy
 5 x 4 y  x 4 ; y (0)  1
dx
dy
 2 y  x 2 ; y (0)  1
dx

0

2.6 EXACT EQUATIONS
2.6 EXACT EQUATIONS
 Solutions
c)
Integrate w.r.t x :
e)
General solution :
2.6 EXACT EQUATIONS
EXACT EQUATIONS
Example : Solve [e y  cos( x  y)  2 x]dx  [ xe y  cos( x  y)  1]dy  0
Example : Solve the differential equation
sin xdy  ( y cos x  x sin x)dx  0
2
2
Example : Solve ( x  y) dx  (2 xy  x  1)dy  0; y(1)  1
Example : Solve y cosh( xy )dx  [ x cosh( xy )  y ]dy  0
where y (0)  20
2.6 EXACT EQUATIONS
Exercises 2.6
Solve the given differential equations.
a)
2 xydx   x 2  y 2  dy  0
b)
(e2 x  y )dx   e  y  x  dy  0
c)
 cos x cos y  2x  dx  sin x sin y  2 y  dy  0
d)
(2 xy  3x 2 )dx  ( x 2  1)dy  0 ; y(0)  1
e)
f)

1
2
 tan y  2  dx   x sec y   dy  0 ; y(0)  1
y

 e x y  1 dx   e x  1 dy  0 ; y(0)  2
2.7 APPLICATIONS OF 1ST DE –
NEWTON’S LAW OF COOLING
2.7 APPLICATIONS OF 1ST DE –
NEWTON’S LAW OF COOLING
 Example : A pie is removed from an oven with
temperature of 350o F and placed to cool
o
in a room with temperature of 75 F . In 15
minutes, the pie has a temperature of 150o F .
Determine the time required to cool the pie
o
to a temperature of 80 F .
 Example : The temperature of a dead body when it was
found at 3 o’clock in the morning is 85o F. The
o
surrounding temperature at that time was 68 F .
After two hours, the temperature of the dead
body decreased to 74o F . Assuming that the
normal body temperature is 98.6o F , determine the
time of murdered.
2.7 APPLICATIONS OF 1ST DE – NEWTON’S
LAW OF COOLING
 Exercises 2.6
a) The oil is heated to 60o C . It cools to 50o C after 6 minutes. Find
the time taken by the oil to cool from 50o C to 40o C .
(Surrounding temperature Ts  25C )
o
b) Robin heats the water to 70 C . He waits for 10 minutes. How
much would be the temperature if k  0.056 per min and
surrounding temperature is 27 o C .
c) Suppose that a corpse was discovered in a motel room at
midnight and its temperature of 80o F . The temperature of the
room is kept constant at 60o F . Two hours later the temperature
of the corpse dropped to 75o F . Find the time of death.
d) A pot of soup start at a temperature of 373K , and the
surrounding temperature is 293K . If the cooling constant is
k  0.001501/ s , what will the temperature of the pot of soup be
after 20 minutes.