Suppose we wish to solve the second-order homogeneous differential equation ay″ + by′ + cy = 0, (3)
where a, b, and c are constants. To solve Equation (3), we seek a function which when multiplied by a constant and added to a constant times its first derivative plus a constant times its second derivative sums identically to zero.
One function that behaves this way is the exponential function y = e^rx
, when r is a constant.
This is how my textbook proceeds to solve the equation (3) and it works, but is y = e^rx the only function that can solve (3) ? Then why does everybody use y=e^rx ?