1) Linear second-order differential equations have the form (1) a(x) y" + b(x) y' + c(x) y+ d(x) = 0. If d(x)=0, then we say that the equation is homogeneous. This means that y(x) = 0 is a solution. (Note that it is not the only solution.) This is called the trivial solution. a) Show that if d(x) = 0, then y(x) = 0 is a solution. (5 points) This problem still may be quite difficult to solve, so next let us assume that each function of x is a constant. In this case our equation becomes (2) ay" + by'+ cy = 0. Since this says that constant multiples of derivatives and the function add to zero, it is natural to believe our solution might be an exponential function. We seek then a solution of the form y = e b) Show that by assuming y = e" we are led to the equation (3) an + ba+ c= 0. (10 points) Our solution then becomes y = Cell + C,ez where 2 and 2, are solutions to equation (3). it involves solving a related