Differential Equations

48
4

4.1: CONSTRUCT THE DIFFERENTIAL EQUATIONS

4.1.1: Identify Type Of Differential Equations

- Order The number of the highest derivative in a differential equation. A differential equation of order 1 is called first order; order 2
second order, etc.
Example:

i.
2
1
1
y
x
dx
dy
+
+
= first order differential equation

ii. x y
dx
dy
x sin
2
+ first order differential equation

iii.
2
2
2
2 4 x y
dx
dy
dx
y d
= + second order differential equation

- Degree The power of the highest order derivative in
the equation. A differential equation of degree is called first degree, second degree, etc.
Example:

i. x xy
dx
dy
xy
2
= first order differential equation with first degree.

ii.
2
2
1 |
.
|
\
|
= +
dx
dy
x y first order differential equation with second degree.

iii. x y
dx
dy
dx
y d
2 cos 10 2
2
2
2
= +
|
|
.
|
\
|
second order differential equation with second degree.

DIFFERENTIAL EQUATIONS
Try this!
Determine order and degree for below equations:
a)
3
2
4

x
x
dx
dy +
= |
.
|
\
|

b) 0 sin
2
= + t
dt
ds
t
c) 0 2 4
2
2
= + + xy
dx
dy
dx
y d
x
d) 0 2 3
2
2
2
= + +
|
|
.
|
\
|
y
dx
dy
dx
y d

49

First order differential equation
4.1.2: Construct The Differential Equation

- Example (a):
Construct the differential equation for y = A sin 2x

Solution:

Step 1: Write down the question as the 1
st
equation

y = A sin 2x

Step 2: Differentiate
for the 1
st
equation (make it as 2
nd
equation)

y = A sin 2x
= 2A cos 2x

Step 3: For the 3
rd
equation, properly arrange the constant.

= 2A cos 2x

)

Step 4: Substitute 3
rd
equation into 1
st
equation and simplify the final calculation.

y = A sin 2x

)
(

- Example (b):
Construct the differential equation for y = Cx
3
+ x
4

Solution:

Step 1: y = Cx
3
+ x
4

Step 2:
= 3Cx
2
+ 4x
3

Step 3:

1
2
3
1
2
3

50

First order
differential equation
Step 4: y = Cx
3
+ x
4

(

- Example (c):
Construct the differential equation for y = Ax
2
Bx + x

Solution:

Step 1: Write down the question as the 1
st
equation

y = Ax
2
Bx + x

Step 2: Differentiate
for the 1
st
equation (make it as 2
nd
equation).

y = Ax
2
Bx + x

= 2Ax B + 1

Step 3: If the differentiation still have 2 constants, do the second order differentiation (make it as 3
rd
equation).

Step 4: For the 3
rd
equation, properly arrange the constant (make it as 4
th
equation).

)

th
equation into 2
nd
equation.

= 2Ax B + 1

*(
) (
)+

) (
)

Step 5: Arrange the second constant properly (make it as 5
th
equation).
(
) (

1
2
4
3
5

51

Second order differential equation
Step 6: Substitute both 4
th
and 5
th
equation into 1
st
equation.
y = Ax
2
Bx + x
*
)+
*(
+
=
) (
) (
) () (
) (
) () (
) () (
)

4.2: FIRST ORDER DIFFERENTIAL EQUATIONS

- There are 4 types of first order differential equations:
o Direct integration
o Separable variables
o Homogenous equation
o Linear equation (integrating factors)

4.2.1: Direct Integration
- Form of
()
- Example (d):
Solve below differential equation

Solution:

Step 1: Solve using direct integration

Try this!
Construct the differential equation for:
a) y = A cos x + B sin x
b) y = Ax
2
+ 3
c) y
2
= 5Ax
d) y = Dx
2
+ Ex

52

4.2.2: Separable Variables

- Form of
()()

()
()

- Example (e):

Solution:

Step 1: Separate two variables with x on the right and y on the left.

Step 2: Solve the integral

tan y = tan x + c

4.2.3: Homogenous Equation

- Form of substitution
- Example (f):
(

Solution:

Step 1: Separate the equation

Step 2: Substitution

)

rd
equation into 1
st
equation
(

1
2
3

53

Step 4: Replace 2
nd
equation into Step 3
(
)
()()
()

()

Step 5: Separate the variables with x and v on different sides


]

Step 7: Replace

)+

54

4.2.4: Linear Equation (Integrating Factors)

- Form of

- Example (g):
( )
( )

Solution:

Step 1: Simplify
) (
)
()
) ( )
) ( )

Step 2: Identify P and Q

( )

Step 3: Integrating factor of P

()

()

()

-ln x = ln x
-1

e
-ln (x-2)
= e
(x-2)-1

e
ln F
= F
= (x-2)
-1

FP =

55

Step 4: Substitute into equation

( )
( )


( )

Step 6: Simplify y

( )
( )
( ) ( )

( )

( )

Try this!
Solve below differential equation:
a) (
) (

b) ( ) (
) ( )

c) (

d) (

Differential Equations

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