SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS

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.

nd-Order ODE - 8 2.2 Linear Independence Two functions, y 1 (x) and y 2 (x), are linearly independent on an interval [x 0, x 1] whenever the relation c 1 y 1 (x) + c 2 y 2 (x) = 0 for all x in the interval implies that c 1 = c 2 = 0. Otherwise, they are linearly dependent. There is an easier way to see if two functions y 1 and y 2

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  Linear, Order, Order linear

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