Transcription of SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS
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2nd- order ODE - 1 CHAPTER 2 SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS 1 homogeneous LINEAR EQUATIONS of the second order LINEAR DIFFERENTIAL Equation of the second order y'' + p(x) y' + q(x) y = r(x) LINEAR where p(x), q(x): coefficients of the equation if r(x) = 0 homogeneous r(x) 0 nonhomogeneous p(x), q(x) are constants constant coefficients 2nd- order ODE - 2 [Example] (i) ( 1 x2 ) y'' 2 x y' + 6 y = 0 y'' 2 x 1 x2 y' + 6 1 x2 y = 0 homogeneousvariable coefficientslinear (ii) y'' + 4 y' + 3 y = ex nonhomogeneousconstant coefficientslinear (iii) y'' y + y' = 0 nonlinear (iv) y'' + (sin x) y' + y = 0 LINEAR , homogeneous ,variable coefficients 2nd- order ODE - 3 second order DIFFERENTIAL EQUATIONS Reducible to the First order Case I.
Consider the second order homogeneous linear differential equa-tion: y'' + p(x) y' + q(x) y = 0 where p(x) and q(x) are continuous functions, then (1) Two linearly independent solutions of the equation can always be found. (2) Let y 1 (x) and y 2 (x) be any two solutions of the homogeneous equa-tion, then any linear combination of them (i.e., c ...
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DIFFERENTIAL, Differential equation, Second order homogeneous, Equation, Order, Homogeneous equation, Second Order, Homogeneous Differential, Second Order Differential Equation Non Homogeneous, Homogeneous, Schrödinger Equation in One Dimension, Second, Homogeneous differential equation, Order differential, ORDINARY DIFFERENTIAL EQUATIONS