pertemuan 9 persamaan diferensial
PERSAMAAN DIFERENSIAL
Pertemuan 9, PD Bernouli
Nikenasih Binatari
[email protected]
TAKWA, MANDIRI, CENDEKIA
http://fmipa.uny.ac.id
PRESENTATION PARTS
1
2
3
• Review PD Homogen
• Review PD Linear
• PD Bernoulli
TAKWA, MANDIRI, CENDEKIA
http://fmipa.uny.ac.id
Review PD Homogen
Definisi homogen
A function
of two variables x and y is said
homogeneous if it satisfies
� ,� =
,
for any number t.
TAKWA, MANDIRI, CENDEKIA
http://fmipa.uny.ac.id
Definition
A function M of two variables x and y is said to be
homogenous of degree-n if it satisfies
� , � = ��
,
TAKWA, MANDIRI, CENDEKIA
http://fmipa.uny.ac.id
Definition
A first differential equation
,
+
,
=
or
=
,
is called homogeneous differential equation if M
and N are homogeneous functions of the same
degree n or f is a homogenous function.
TAKWA, MANDIRI, CENDEKIA
http://fmipa.uny.ac.id
Menyelesaikan PD Homogen
PD Homogen (dalam x dan y)
Transform y = vx
PD separabel (dalam v dan x)
Selesaikan PD separabel
Inverse transform v = y/x
TAKWA, MANDIRI, CENDEKIA
http://fmipa.uny.ac.id
PD Linear
Definition
A first-order ordinary differential equation is linear
in the dependent variable y and the independent
variable x if it is, or can be, written in the form
+
TAKWA, MANDIRI, CENDEKIA
=
http://fmipa.uny.ac.id
Menyelesaikan PD Linear
Suppose the integrating factor
�
= �
�
The general solution of linear differential equation
as follows
= �−
TAKWA, MANDIRI, CENDEKIA
�
+
http://fmipa.uny.ac.id
PD Bernoulli
Definition
An equation of the form
+
=
�
is called a Bernoulli differential equation.
TAKWA, MANDIRI, CENDEKIA
http://fmipa.uny.ac.id
Theorem
Suppose n ≠ 0 or n ≠ 1, then the transformation � =
−� reduces the Bernoulli equation
+
into a linear equation in v.
TAKWA, MANDIRI, CENDEKIA
=
�
http://fmipa.uny.ac.id
Example
Solve the differential equation below
+
TAKWA, MANDIRI, CENDEKIA
=
http://fmipa.uny.ac.id
Answer :
To solve the differential equations above, first we
have to divided both side with
. Here we get,
Suppose � =
−
− �
�
−
−
+
then
−
��
�
−
=−
. By here, we get
�
+ − �=
TAKWA, MANDIRI, CENDEKIA
=
−
or else
��
�
=
http://fmipa.uny.ac.id
To solve the equation, we use integrating factor
�
Then the solution is
�
=
� = �−
�=
=
TAKWA, MANDIRI, CENDEKIA
−
ln
=
�
−
+
=
+
−
=
+
→�=
+
−
−
+
http://fmipa.uny.ac.id
Solve the following differential equations.
.
−
.
+
.
.
+
+
=
=
=
=
TAKWA, MANDIRI, CENDEKIA
http://fmipa.uny.ac.id
Referensi :
1. Ross, L. Differential Equations. John Wiley &
Sons.
TAKWA, MANDIRI, CENDEKIA
http://fmipa.uny.ac.id
THANK YOU
Nikenasih Binatari
[email protected]
Karangmalang Sleman Yogyakarta
TAKWA, MANDIRI, CENDEKIA
http://fmipa.uny.ac.id
Pertemuan 9, PD Bernouli
Nikenasih Binatari
[email protected]
TAKWA, MANDIRI, CENDEKIA
http://fmipa.uny.ac.id
PRESENTATION PARTS
1
2
3
• Review PD Homogen
• Review PD Linear
• PD Bernoulli
TAKWA, MANDIRI, CENDEKIA
http://fmipa.uny.ac.id
Review PD Homogen
Definisi homogen
A function
of two variables x and y is said
homogeneous if it satisfies
� ,� =
,
for any number t.
TAKWA, MANDIRI, CENDEKIA
http://fmipa.uny.ac.id
Definition
A function M of two variables x and y is said to be
homogenous of degree-n if it satisfies
� , � = ��
,
TAKWA, MANDIRI, CENDEKIA
http://fmipa.uny.ac.id
Definition
A first differential equation
,
+
,
=
or
=
,
is called homogeneous differential equation if M
and N are homogeneous functions of the same
degree n or f is a homogenous function.
TAKWA, MANDIRI, CENDEKIA
http://fmipa.uny.ac.id
Menyelesaikan PD Homogen
PD Homogen (dalam x dan y)
Transform y = vx
PD separabel (dalam v dan x)
Selesaikan PD separabel
Inverse transform v = y/x
TAKWA, MANDIRI, CENDEKIA
http://fmipa.uny.ac.id
PD Linear
Definition
A first-order ordinary differential equation is linear
in the dependent variable y and the independent
variable x if it is, or can be, written in the form
+
TAKWA, MANDIRI, CENDEKIA
=
http://fmipa.uny.ac.id
Menyelesaikan PD Linear
Suppose the integrating factor
�
= �
�
The general solution of linear differential equation
as follows
= �−
TAKWA, MANDIRI, CENDEKIA
�
+
http://fmipa.uny.ac.id
PD Bernoulli
Definition
An equation of the form
+
=
�
is called a Bernoulli differential equation.
TAKWA, MANDIRI, CENDEKIA
http://fmipa.uny.ac.id
Theorem
Suppose n ≠ 0 or n ≠ 1, then the transformation � =
−� reduces the Bernoulli equation
+
into a linear equation in v.
TAKWA, MANDIRI, CENDEKIA
=
�
http://fmipa.uny.ac.id
Example
Solve the differential equation below
+
TAKWA, MANDIRI, CENDEKIA
=
http://fmipa.uny.ac.id
Answer :
To solve the differential equations above, first we
have to divided both side with
. Here we get,
Suppose � =
−
− �
�
−
−
+
then
−
��
�
−
=−
. By here, we get
�
+ − �=
TAKWA, MANDIRI, CENDEKIA
=
−
or else
��
�
=
http://fmipa.uny.ac.id
To solve the equation, we use integrating factor
�
Then the solution is
�
=
� = �−
�=
=
TAKWA, MANDIRI, CENDEKIA
−
ln
=
�
−
+
=
+
−
=
+
→�=
+
−
−
+
http://fmipa.uny.ac.id
Solve the following differential equations.
.
−
.
+
.
.
+
+
=
=
=
=
TAKWA, MANDIRI, CENDEKIA
http://fmipa.uny.ac.id
Referensi :
1. Ross, L. Differential Equations. John Wiley &
Sons.
TAKWA, MANDIRI, CENDEKIA
http://fmipa.uny.ac.id
THANK YOU
Nikenasih Binatari
[email protected]
Karangmalang Sleman Yogyakarta
TAKWA, MANDIRI, CENDEKIA
http://fmipa.uny.ac.id