3. Differential Equation - Homogeneous Differential- Lecture.2024.pptx

Differential Equation

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

2. Method 2: Homogeneous Differential Equation 29 August 2024 Prepared By: Engr. Ma. Cristina Macawile 2 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. 29 August 2024 Prepared By: Engr. Ma. Cristina Macawile 3 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. 29 August 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 , 2 nd degree    

5. 29 August 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 , 4 th degree    

6. 29 August 2024 Prepared By: Engr. Ma. Cristina Macawile 6 Determine if homogeneous or nonhomogeneous. If homogeneous, determine the degree.   F (x, y) =     2. F ( x,y ) =    

7. 29 August 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  

8. 29 August 2024 Prepared By: Engr. Ma. Cristina Macawile 8 Step 3: Simplify and combine similar terms. ( = 0   =0    

9. 29 August 2024 Prepared By: Engr. Ma. Cristina Macawile 9 Step 4: Perform variable separable method. Multiply the equation by .                   (      

10. 29 August 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. 29 August 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 = ,  

12. 29 August 2024 Prepared By: Engr. Ma. Cristina Macawile 12 Step 3: Simplify and combine similar terms.       Step 4: Perform variable separable method. Multiply the equation by .        

13. 29 August 2024 Prepared By: Engr. Ma. Cristina Macawile 13 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. 29 August 2024 Prepared By: Engr. Joshua Hernandez 14

15. 29 August 2024 15 Prepared By: Engr. Ma Cristina Macawile