The document provides an explanation of homogeneous differential equations, detailing their standard form and characteristics. It includes methods for determining homogeneity and examples demonstrating the steps for solving both general and particular solutions. Key concepts covered include identifying the degree of homogeneity and applying the variable separable method.

1. Solutions of First Order, First Degree Ordinary Differential Equation
Homogeneous Differential Equation
Lecture

2. September 6, 2024 2
Method 2: Homogeneous Differential Equation
Prepared By: Engr. Ma. Cristina Macawile
Standard Form
M (x, y) dx + N (x,y) dy = 0
where M (x,y) and N(x,y) dy are homogeneous functions in the same
degree
The equations can be reduced to variable separable by using the
following substitutions
Let

3. September 6, 2024 3
Prepared By: Engr. Ma. Cristina Macawile
Definition of a homogeneous function
F (x,y) is said to homogeneous if ,
F (x , y) = F k
(x,y)
That is if x and y are simultaneously replaced by x
and y , the original function multiplied by k
results.
k – being the degree of homogeneity

4. September 6, 2024 Prepared By: Engr. Ma. Cristina Macawile 4
Direction: Determine if the function is homogeneous. Determine the
degree of homogeneity
Sample 1 : F (x, y) =
Step 1: Replace all ‘x’ by x and ‘y’ by y
F (x , y) =
=
Step 2: Factor out the ‘ ‘
F (x , y) = (
F k
(x,y) = (
Step 3: Identify the value of k. F (x,y) is said to homogeneous if ,
F (x , y) = F k
(x,y)
k = 2 , the function is homogeneous , 2nd
degree

5. September 6, 2024 Prepared By: Engr. Ma. Cristina Macawile 5
Sample 2. F (x, y) =
Step 1: Replace all ‘x’ by x and ‘y’ by y
F (x , y) =
=
Step 2: Factor out the 
F k
(x,y) =
Step 3: Identify the value of k. F (x,y) is said to homogeneous if ,
F (x , y) = F k
(x,y)
k = 4 , the function is homogeneous , 4th
degree

6. September 6, 2024 Prepared By: Engr. Ma. Cristina Macawile 6
Determine if homogeneous or nonhomogeneous. If homogeneous,
determine the degree.
1. F (x, y) =
2. F (x,y) =

7. September 6, 2024 Prepared By: Engr. Ma. Cristina Macawile 7
Homogeneous Differential Equation
Sample 1: Solve for general solution
Step 1: Check if homogeneous, if yes proceed.
Homogeneous, degree 2
Step 2: Replace x = , or
y = ,
( = 0

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Step 3: Simplify and combine similar terms.
( = 0
=0

9. September 6, 2024 Prepared By: Engr. Ma. Cristina Macawile 9
Step 4: Perform variable separable method. Multiply the equation by .
|− 𝑣 +5
𝑣 +2
(
7
−1+
7
𝑣 +2

10. September 6, 2024 Prepared By: Engr. Ma. Cristina Macawile 10
Step 5: Back substitute the . Write the general solution
This is from the previous representation, x = .
General Solution

11. September 6, 2024 Prepared By: Engr. Ma. Cristina Macawile 11
Sample 2: Solve for general solution and particular solution.
x = 1, y =
Step 1: Check if homogeneous, if yes proceed.
Homogeneous, degree 1
Step 2: Replace y = , or
x = ,
For this example we choose, y = ,

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Step 3: Simplify and combine similar terms.
Step 4: Perform variable separable method. Multiply the equation by .

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Step 5: Back substitute the to solve for general solution . This is from the
previous representation, y = . Write the general solution
Step 6: Substitute values of x and y to solve for particular solution
General Solution
Particular Solution

14. END OF PRESENTATION
ONE HEART.
ONE COMMITMENT.
ONE LIFE.
1719 – 2019
#300LaSalle
September 6, 2024 Prepared By: Engr. Joshua Hernandez 14

15. The Lasallian Prayer
“I will continue, O my God,
to do all my actions for the love of Thee.”
Saint John Baptist de La Salle, pray for us.
Live Jesus in our hearts, forever.
September 6, 2024 Prepared By: Engr. Ma Cristina Macawile 15