Higher order linear differential Equation UG BSc Part2 Dr Md Ataur Rahman Guest Faculty Department of Mathematics M L Arya College Kasba PURNEA UNIVERSITY PURNIA Contents ID: 1030908
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1. Differential EquationLecture-21Higher order linear differential EquationUG (B.Sc., Part-2)Dr. Md. Ataur RahmanGuest FacultyDepartment of MathematicsM. L. Arya, College, KasbaPURNEA UNIVERSITY, PURNIA
2. Contentslinear differential equation of higher orderlinear differential equation with constant co-efficientslinear differential equation of second orderComplementary function (C.F)Formation of Auxiliary equationParticular Integral (P. I).Problems
3. Higher order linear differential EquationA differential equation of the formWhere are constants or functions of x alone. is called linear differential equation of nth order.Remark:- If are constants and Q is a function of x only or constant. Then (1) is called linear differential equation of nth order with constants co-efficients.
4. Linear differential Equation with Constant co-efficients A differential equation of the formWhere are constants and Q is a constant or function of x only. is called linear differential equation of nth order with constants co-efficients.
5. Solution of (2)Then general solution or complete solution of (2) is y ═ C.F+P.I…..(3)Where C.F═ Complementary function and P.I ═Particular IntegralRemarks:- If Q ═0, Then general solution of (2) is y ═ C.F…..(4)
6. Formation of Auxiliary EquationFrom(2),Then auxiliary equation of (5) is Let be the roots of Eq. (6).
7. Method of finding complementary functionC.F depends on the nature of the roots of the auxiliary Eq.(6) according asThe roots are real and distinct (different)The roots are real and equal (repeated)The roots are conjugate complex and distinct.The roots are conjugate complex and repeated
8. When the roots are real and distincti.e. all are distinct.ThenExample1. SolveSolution:- Given Eq. is The auxiliary Eq. is Then general solution of (1) is
9. When the roots are real and repeatedSuppose k roots are equal and remaining distinct.i .e. ThenExample1. SolveSolution:- Given Eq. is The auxiliary Eq. is Then general solution of (1) is
10. When the roots are complex conjugate and distincti.e. ThenExample1. SolveSolution:- Given Eq. is The auxiliary Eq. is Then general solution of (1) is