Solving Second Order Linear Homogeneous Differential Equations

Any differential equation of the form is a second order differential equations and there is a standard technique for solving any equation of this sort. We assume a solution of the form and substitute this into the equation. We extract the non zero factor – since no exponential is zero for any finite x - to obtain a quadratic equation. We solve this equation obtain solutions and and then the general solution is and may be evaluated given suitable boundary conditions, for example Example: Solve the equation (1) subject to and at  Substitution of the above expressions into (1) gives We can factor out the nonzero to obtain Because is non zero we can divide by it to obtain and we factorise this expression to obtain and solve to obtain and The general solution is then We now have to find  when implies (2) at implies (3)

(2)+(3) gives then from (2) The solution is Example: Solve the equation (1) subject to when and at  Substitution of the above expressions into (1) gives We can move to the left hand side and factor out the nonzero to obtain Because is non zero we can divide by it to obtain and we factorise this expression to obtain and solve to obtain and The general solution is then We now have to find  when implies (2) at implies (3) (2)-(3) gives then from (2) The solution is  