1. Differential Equations MTH 242 Lecture # 09 Dr. Manshoor Ahmed

2. Summary (Recall) Higher order linear differential equation. Homogeneous and non-homogeneous equations with constant coefficients. Initial value problem (IVP) and it’s solution. Existence and Uniqueness of Solutions. Boundary value problem (BVP) and it’s solution. Linear independence and dependence of functions. Wronskian of a set of functions.

3. Solution of Higher Order Linear Equation

4. Preliminary Theory A linear nth order equation of the form is said to be homogeneous. A linear nth order equation of the form, where g(x) is not identically zero, is said to be non-homogeneous.

5. Solution of Higher Order Linear Equation For solution of non-homogeneous DE, we first solve the associated homogeneous differential equation. So, first we learn how to solve the homogeneous DE. As function y=f(x) that satisfies the associated homogeneous equation is called solution of the differential equation.

6. Solution of Higher Order Linear Equation Theorem: (Superposition principle Homogeneous equations) Let y 1,y 2,…,y k be solutions of homogeneous linear nth order differential equation on an interval I then the linear combination, where the c i, i = 1,2,…,k are arbitrary constants, is also a solution on the interval. Remarks: i. A constant multiple y=c 1 y 1 (x) of a solution y 1 (x) of a homogeneous DE is also a solution. ii. A homogeneous linear differential equation always possesses the trivial solution y = 0.

7. Solution of Higher Order Linear Equation  The superposition principle is valid of linear DE and it does not hold in case of non-linear DE. Example 1: The functions y 1 (x) = x 2 and y 1 (x) = x 2 lnx are both solutions of homogeneous equation on Then by Superposition principle is also the solution of the DE on the interval.

8. Solution of Higher Order Linear Equation Example 2 The functions all satisfy the homogeneous DE on. Thus are all solutions of the given DE. Now suppose that

9. Solution of Higher Order Linear Equation Then, Therefore, Thus, is a solution the DE.

10. Solution of Higher Order Linear Equation Linearly independent solutions Theorem: Let y 1,y 2,…, y n be solutions of homogeneous linear nth order DE on an interval I. Then the set of solutions is linearly independent on I if and only if for every x in the interval.

11. Solution of Higher Order Linear Equation In other words The solutions y 1,y 2,…, y n are linearly dependent if and only if For example, we consider a second order homogeneous linear DE and Suppose that y 1,y 2 are two solutions on an interval I. Then either Or

12. Solution of Higher Order Linear Equation To verify this we write the equation as Now Differentiating w.r.t x, we have Since y 1 and y 2 are solutions of the DE

13. Solution of Higher Order Linear Equation Therefore, Multiplying 1 st equation by y 2 and 2 nd by y 1 then we have Subtracting the two equations we have: Or

14. Solution of Higher Order Linear Equation This is a linear 1 st order differential equation in W, whose solution is Therefore  If then the solutions are linearly independent on I.  If then the solutions are linearly dependent on I.

15. Solution of Higher Order Linear Equation Fundamental set of solutions Any set y 1,y 2,…, y n of n linearly independent solutions of the homogeneous linear nth order DE (3) on an interval I is said to be a Fundamental set of solutions on the interval. General Solution of homogeneous equations: Let y 1,y 2,…, y n be fundamental set of solutions of the homogeneous linear nth order differential equation (3) on an interval I, then the general solution of the equation on the interval is defined to be where the c i, i = 1,2,…,n are arbitrary constants.

16. Solution of Higher Order Linear Equation Example 1 The functions y 1 (x) = e 3x,y 2 (x) = e 2x and y 3 (x) = e 3x satisfy DE Since, For x, y 1, y 2, y 3 every real value form a Fundamental set of solutions on and is the general solution on the given interval.

17. Solution of Higher Order Linear Equation

18. The general solution of the differential equation is If we take,. We get Hence, the particular solution has been obtained from the general solution.

19. Solution of Higher Order Linear Equation Solution of Non-Homogeneous Equations A function y p that satisfies the non-homogeneous differential equation and is free of parameters is called the particular solution of the differential equation Example 1 Suppose that Then,

20. Solution of Higher Order Linear Equation Therefore, Hence, is a particular solution of the differential equation

21. Solution of Higher Order Linear Equation Complementary function The general solution of the homogeneous linear differential equation is known as the complementary function for the non-homogeneous linear differential equation.

22. Solution of Higher Order Linear Equation General Solution Non-Homogeneous equations Suppose that The particular solution of the non-homogeneous equation is y p. The complementary function of the non-homogeneous differential equation is

23. Solution of Higher Order Linear Equation Then general solution of the non-homogeneous equation on the interval ‘I’is given by: Or Hence General Solution = Complementary Solution + Any Particular Solution

24. Solution of Higher Order Linear Equation Example Let is a particular solution of the equation Then the complementary function of the associated homogeneous equation is

25. Solution of Higher Order Linear Equation So the general solution is Implies

26. Solution of Higher Order Linear Equation Superposition Principle Non-Homogeneous Equations Let be ‘k’ particular solutions of a linear nth order differential equation (4) on an interval I corresponding to ‘k’ distinct functions g 1,g 2,…, g k that is denotes a particular solution of the corresponding differential equation where i = 1,2,…,k.Then is a particular solution of

27. Solution of Higher Order Linear Equation Example Consider the differential equation Suppose that Then, Therefore, is a particular solution of the non-homogenous differential equation

28. Solution of Higher Order Linear Equation Similarly, it can be verified that are particular solutions of the equations: And respectively. Hence, is a particular solution of the differential equation

29. Exercises Verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. Form the general solution.

32. Exercises Verify that the given two-parameter family of functions is the general solution of the nonhomogeneous differential equation on the indicated interval.

34. Summary Solution of Higher order linear differential equation. Solution of homogeneous equations. Superposition principle. Linearly independent Solutions. Fundamental set of solutions, general solution. Solution of non-homogeneous equations. Complementary function and particular solution.