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

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

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

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

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

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Second-Order ODE--- y’’ + Ay’ + By = 0 ( 常係數 ) Case 3 : Two distinct, complex values for r (when A 2 -4B < 0) Any linear combination of solutions is also a solution ! The general solution : Where c 1 and c 2 are arbitrary constants Chapter 2 Linear Differential Equations of Second and Higher Order

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

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

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

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

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

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung 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 c 1 and c 2 are arbitrary constants Euler - Cauchy equation Chapter 2 Linear Differential Equations of Second and Higher Order

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Linear Nonhomogeneous Second-Order ODE Let y 1 (x) and y 2 (x) are solutions of y’’+ P(x)y’ + Q(x)y = 0 on the interval J, Let y p (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) = c 1 y 1 (x) + c 2 y 2 (x) + y p (x) How to find the particular solution y p (x) of y’’+ P(x)y’ + Q(x)y = F(x) ? Chapter 2 Linear Differential Equations of Second and Higher Order

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

Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Linear Nonhomogeneous Second-Order ODE Method 2 : Variation of Parameters --- for P(x) and Q(x) need not to be constant Let y 1 (x) and y 2 (x) are solutions of y’’+ P(x)y’ + Q(x)y = 0 on the interval J, Try y p (x) = u(x)y 1 (x) + v(x)y 2 (x) Impose the condition : y p (x) = u(x)y 1 (x) + v(x)y 2 (x) Chapter 2 Linear Differential Equations of Second and Higher Order

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

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

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

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

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

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