MMLLLLLLLLL

1. lecture 2
Assist. Lect. Hiba Abdul –Kareem

2. Homogeneous Differential Equation
•A differential equation containing a homogeneous function
is called a homogeneous differential equation.
•A differential equation of the form :
𝒅𝒚
𝒅𝒙
=
𝒇(𝒙,𝒚)
𝒈(𝒙,𝒚)
called homogeneous differential equation if f (x, y) and g(x, y)
are homogeneous functions

3. Homogeneous Differential Equation
• A differential equation of the form :
𝒅𝒚
𝒅𝒙
=
𝒇(𝒙,𝒚)
𝒈(𝒙,𝒚)
• Example :
𝒅𝒚
𝒅𝒙
=
𝒙𝟑+𝒚𝟑
𝟑𝒙𝟐𝒚
Separable Differential Equation
• A differential equation of the form :
𝒅𝒚
𝒅𝒙
=
𝒇(𝒙)
𝒈(𝒚)
or
• Example :
𝒅𝒚
𝒅𝒙
=
𝒙𝟑
𝒚𝟐
The homogeneous differential equation does not have a constant term within
the equation

4. Homogeneous Function
• The function f(x, y) is called a homogeneous function if :
𝒇(λx, λy)=𝝀𝒏
𝒇(x, y), for any non zero constant λ.
• Example :Determine whether the function is homogeneous or not?
𝒇 𝒙, 𝒚 = 𝒙𝟑+𝒙𝟐y+𝒚𝟑
𝒇 λ𝒙, λ𝒚 = (λ𝒙)𝟑
+(λ𝒙)𝟐
(λ y)+(λ𝒚)𝟑
𝒇 λ𝒙, λ𝒚 = λ𝟑
(𝒙𝟑
+𝒙𝟐
y+𝒚𝟑
)
the function is homogeneous
𝒇 𝒙, 𝒚 = 𝒙𝒚+𝒔𝒊𝒏 𝒙𝒚
𝒇 λ𝒙, λ𝒚 =(λx)(λy)+sin(λ𝟐xy)
𝒇 λ𝒙, λ𝒚 =λ𝟐xy+sin(λ𝟐xy)
the function is non -homogeneous

5. Classwork
Show that the function 𝒇 𝒙, 𝒚 =
𝒙+𝟐𝒚
𝒙−𝒚
is a homogeneous
function.

6. Classwork
Show that the function 𝒇 𝒙, 𝒚 =
𝒙+𝟐𝒚
𝒙−𝒚
is a homogeneous
function.

7. Homogeneous ODE
• The differential equation may not be separable, but at the same time, it
can be converted into a separable equation using some transformations.
So, the homogeneous differential equation are is the one which can be
written in this form

8. Homogeneous ODE
• Example: Determine whether the differential equation is
homogeneous or not:
Homogeneous

9. Classwork
• Example: Determine whether the differential equation is
homogeneous or not:

10. Classwork
• Example: Determine whether the differential equation is
homogeneous or not:
Homogeneous

11. Solution of Homogeneous ODE
‫كل‬ ‫بدل‬ ‫نعوض‬
V
‫ب‬
𝒚
𝒙
‫ث‬ ‫المعادلتين‬ ‫ربط‬ ‫يتم‬
‫م‬
‫الطرفين‬ ‫تكامل‬ ‫ناخذ‬
‫نشتق‬
y=vx
‫ل‬ ‫بالنسبة‬
x
‫حيث‬
𝒅𝒚
𝒅𝒙
=
𝒅𝒚
𝒅𝒙
𝐱+v
‫متغير‬ ‫نفرض‬
‫جديد‬
(
v
)
‫حيث‬
:
𝒗 =
𝒚
𝒙
‫المعادلة‬ ‫نجعل‬
‫بالصيغة‬
:
𝒅𝒚
𝒅𝒙
=f(
𝒚
𝒙
)

12. Solution of Homogeneous ODE

13. Solution of Homogeneous ODE
Example –Solve
𝒅𝒚
𝒅𝒙
=
𝒙 − 𝒚
𝒙 + 𝒚
Look at the power of x in nominator
So, divide both side by x
Answer:

14. 𝒅𝒚
Replace
𝒅𝒙
= 𝒗 + 𝒙 𝒅𝒗
𝒙
And
𝒚
𝒙
= 𝒗

17. Example – Solve the following differential equation
𝒅𝒚
𝒅𝒙
=
𝒙𝟐 + 𝒚𝟐
𝒙𝒚
Answer:
First, try to write it in the form of
𝑑𝑦
𝑑𝑥
= 𝑓
𝑦
𝑥
11

22. Classwork
•Solve:
(𝟑𝒙 − 𝒚)
𝒅𝒚
𝒅𝒙
=(𝒙 + 𝒚)

25. Homework
•Solve:

𝑑𝑦
𝑑𝑥
=
3𝑦2−𝑥2
2𝑥𝑦

𝑑𝑦
𝑑𝑥
=
𝑦
𝑥
+𝑒
𝑦
𝑥

26. Any
Questions