Homogeneous Linear Differential Equations

Definition, Method of solution and solved problems of Homogeneous Linear Differential Equations

1. HOMOGENEOUS LINEAR DIFFERENTIAL EQUATIONS

2. Definition An equation of the form Where ,…, are constants and X is a function of is called a homogeneous linear differential equation of the nth order .  

3. Method of Solution Reduce the given homogeneous linear differential equation into linear equation with constant coefficients by putting and so on Given equation becomes Take as a trial solution S olve the equation Replace  

4. Prove that the following:   Proof: Let Then or,   Then by chain rule , we get proved  

5. b) We have   Therefore  

6. b) We have  

7. Therefore  

8. Solution: Given equation is which is a linear equation in y  

9. Let be a trial solution of Then we get Then the A.E. (Auxiliary Equation) is  

10. Then C.F. Hence the complete solution is where  

11. Solution: Given equation is which is a linear equation in y  

12. Let be a trial solution of Then we get Then the A.E. (Auxiliary Equation) is  

13. Then C.F. Hence the complete solution is where  

14. Solution: Given that which is a linear equation in y  

15. Let be a trial solution of Then we get Then the A.E. (Auxiliary Equation) is  

16. Then C.F. Hence the complete solution is where  

17. Solution: Given that which is a linear equation in y  

18. Let be a trial solution of Then we get Then the A.E. (Auxiliary Equation) is  

19. Then C.F.  

20. Hence the complete solution is where  

21. Solution: Given equation is which is a linear equation in y  

22. Let be a trial solution of Then we get Then the A.E. (Auxiliary Equation) is Then C.F.  

23. Hence the complete solution is where