Answer Happy

Q2. Consider the third-order linear homogeneous ordinary differential equa- tion with variable coefficients 2 day tar dy d.x2 + Bir + 4y = 0, I>0, dr d.r3 where a, b, are real constants. Show that the substitution t = Inx transforms this equation into an equation with constant coefficients. Now let a = 5,8 = 2,7 = -2, suggest the solution y(t) = emt to obtain the characteristic equation of the transformed ordinary differ- ential equation, and thus solve it, clearly identifying the three linearly independent solutions of the original equation above. Finally, using the substitution y = x show that with the same pa- rameter values for a, , 7 we obtain the characteristic equation for the transformed equation when m is replaced by p. (15 marks