A NOTE ON A DIFFERENTIAL EQUATION

Russell Euler Northwest Missouri State University

If a ≠ b, then two linearly independent solutions of the differential equation

(1)  y′′ − (a + b)y′ + aby = 0

are e^{ax} and e^{bx} and the general solution of (1) is

(2)  y = c_{1}e^{ax} + c_{2}e^{bx} .

When a = b , the two fundamental solutions are y_{1} = e^{ax} and y_{2} = xe^{ax}. This is easy to check but not so easy to motivate, especially y_{2} . The motivation of the form of the general solution in the case of equal roots of the characteristic equation can be accomplished by considering the case when a ≠ b , renaming the constants in equation (2), and considering the limit as b approaches a.

Letting

c_{1} = c_{3} − \frac{c_{4}}{b − a} and c_{2} = \frac{c_{4}}{b − a}

in (2) leads to

(3)  y = c_{3}e^{ax} + c_{4} \frac{e^{bx} − e^{ax}}{b − a}

as the general solution of (1).

Notice that when a = b the second term in equation (3) is of the indeterminate form \frac{0}{0} . So, employing L’Hospital’s rule to compute the limit as b approaches a in (3) yields

y = c_{3}e^{ax} + c_{4}xe^{ax} ,

which is the general solution of (1) when a = b and the technique utilized clearly shows how y_{2} = xe^{ax} arises.