Chapter 2 Linear Differential Equations of Second and Higher Order

1. Chapter 2 Linear Differential Equations of Second and Higher Order Second-Order ODE Linear second order differential equation Initial value problem Homogeneous Superposition principle (Linear principle) If y1(x) and y2(x) are solutions of y’’+ P(x)y’ + Q(x)y = 0 on the interval J, and c1 and c2 are any real numbers, then c1y1(x) + c2y2(x) is also a solution on J If y1(x) and y2(x) are solutions of y’’+ P(x)y’ + Q(x)y = 0 on the interval J, and y2(x) is a constant multiple of y1(x) on an interval J, and vice versa, then we call y1(x) and y2(x) are linearly dependent on J Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

2. Chapter 2 Linear Differential Equations of Second and Higher Order Second-Order ODE Let y1(x) and y2(x) are solutions of y’’+ P(x)y’ + Q(x)y = 0 on the interval J, Let W(y1,y2) = y1(x )y’2(x)- y’1(x )y2(x) then we call y1(x) and y2(x) are linearly independent on J if and only if W(y1,y2)  0 Wronskian Let y1(x) and y2(x) are linearly independent solutions of y’’+ P(x)y’ + Q(x)y = 0 on the interval J, then the expression c1y1(x) + c2y2(x) is the general solution on J Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

3. Chapter 2 Linear Differential Equations of Second and Higher Order Second-Order ODE--- y’’ + Ay’ + By = 0 (常係數) The basic method of solution Characteristic equation (Auxiliary equation) 1.Two distinct, real values for r (when A2-4B > 0) 2.Only one real value for r (when A2-4B = 0) 3.Two distinct, complex values for r (when A2-4B < 0) Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

4. Chapter 2 Linear Differential Equations of Second and Higher Order Second-Order ODE--- y’’ + Ay’ + By = 0 (常係數) Case 1 : Two distinct, real values for r (when A2-4B > 0) The general solution : Where c1 and c2 are arbitrary constants Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

5. Chapter 2 Linear Differential Equations of Second and Higher Order Second-Order ODE--- y’’ + Ay’ + By = 0 (常係數) Case 2 : Only one real value for r (when A2-4B = 0) To find y2(x), we apply a technique called reduction of order Try to produce u(x) such that u(x) y1(x) is a solution. y’’ + Ay’ + By = 0 代入 If we choose c1 = 1 and c2 = 0 The general solution : Where c1 and c2 are arbitrary constants Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

6. Chapter 2 Linear Differential Equations of Second and Higher Order Second-Order ODE--- y’’ + Ay’ + By = 0 (常係數) Case 3 : Two distinct, complex values for r (when A2-4B < 0) Any linear combination of solutions is also a solution ! The general solution : Where c1 and c2 are arbitrary constants Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

7. Chapter 2 Linear Differential Equations of Second and Higher Order Homework Damped oscillations – the mechanical energy of the system diminishes in time Retarding force : (a). Under-damped : (b). Critically damped : (c). Over-damped : Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

8. Chapter 2 Linear Differential Equations of Second and Higher Order Homework The RLC Circuit Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

9. Chapter 2 Linear Differential Equations of Second and Higher Order Euler - Cauchy equation The basic method of solution Characteristic equation 1.two distinct, real values for r (when (A-1)2-4B > 0) 2.Only one real value for r (when (A-1)2-4B = 0) 3.Two distinct, complex values for r (when (A-1)2-4B < 0) Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

10. Chapter 2 Linear Differential Equations of Second and Higher Order Euler - Cauchy equation Case 1 : Two distinct, real values for r (when (A-1)2-4B > 0) The general solution : Where c1 and c2 are arbitrary constants Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

11. Chapter 2 Linear Differential Equations of Second and Higher Order Euler - Cauchy equation Case 2 : Only one real value for r (when (A-1)2-4B = 0) To find y2(x), we apply a technique called reduction of order Try to produce u(x) such that u(x) y1(x) is a solution. 代入 The general solution : Where c1 and c2 are arbitrary constants Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

12. Chapter 2 Linear Differential Equations of Second and Higher Order Euler - Cauchy equation Case 3 : Two distinct, complex values for r (when (A-1)2-4B < 0) Any linear combination of solutions is also a solution ! The general solution : Where c1 and c2 are arbitrary constants Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

13. Chapter 2 Linear Differential Equations of Second and Higher Order Linear Nonhomogeneous Second-Order ODE Let y1(x) and y2(x) are solutions of y’’+ P(x)y’ + Q(x)y = 0 on the interval J, Let yp(x) be any solution of y’’+ P(x)y’ + Q(x)y = F(x) on the interval J then every solution of y’’+ P(x)y’ + Q(x)y = F(x) on J is the form y(x) = c1y1(x) + c2y2(x) + yp(x) How to find the particular solution yp(x) of y’’+ P(x)y’ + Q(x)y = F(x) ? Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

14. Chapter 2 Linear Differential Equations of Second and Higher Order Linear Nonhomogeneous Second-Order ODE Method 1 : Undetermined Coefficients --- for P(x) and Q(x) are constant or or or Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

15. Chapter 2 Linear Differential Equations of Second and Higher Order Linear Nonhomogeneous Second-Order ODE Method 2 : Variation of Parameters --- for P(x) and Q(x) need not to be constant Let y1(x) and y2(x) are solutions of y’’+ P(x)y’ + Q(x)y = 0 on the interval J, Try yp(x) = u(x)y1(x) + v(x)y2(x) Impose the condition : yp(x) = u(x)y1(x) + v(x)y2(x) Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

16. Chapter 2 Linear Differential Equations of Second and Higher Order Forced Oscillation, Resonance If Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

17. Chapter 2 Linear Differential Equations of Second and Higher Order Forced Oscillation, Resonance 令 Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

18. Chapter 2 Linear Differential Equations of Second and Higher Order Forced Oscillation, Resonance Undamped c = 0 Resonance factor Resonance Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

19. Chapter 2 Linear Differential Equations of Second and Higher Order Forced Oscillation, Resonance Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

20. Chapter 2 Linear Differential Equations of Second and Higher Order Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung

21. Chapter 2 Linear Differential Equations of Second and Higher Order Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung