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: F(x, y', y'') = 0 y does not appear explicitly [Example] y'' = y' tanh x [Solution] Set y' = z and dzydx Thus, the DIFFERENTIAL equation becomes first order z' = z tanh x which can be solved by the method of separation of variables dz z = tanh x dx = sinh x cosh x dx or ln|z| = ln|cosh x| + c' z = c1 cosh x or y' = c1 cosh x Again, the above equation can be solved by separation of variables: dy =

nd-Order ODE - 9 2.3 General Solution 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-

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