Question

3. Consider4 the homogenous linear second order differential equation

y′′ − 2y′ + y = 0 (⋆)

(a) Verify that the function y = e^x is a solution of equation (⋆) on the interval (−∞, ∞).

(b) Verify that the function y = xex is a solution of equation (⋆) on the interval (−∞, ∞).

(c) Verify that y = 7e^x + (5xe)^x is a solution of equation (⋆) on the interval (−∞, ∞).

(d) Assume that c and d are any two fixed real numbers. Verify that the function y = (ce)^x + d(xe)^x

is a solution of equation (⋆) on the interval (−∞, ∞).

Note that your answer in part (d) is the most general. Indeed, as was done in question 2(f), show that all results in parts (a) through (c) are immediate consequences of the general result in (d), by using suitable values of the constants c and d.

That is, fill in the blanks below:

Part (a) follows from (d) using the constants c = part (b) follows from (d) using the constants c = part (c) follows from (d) using the constants c =

and d = , and d = , and d = ,

(f) Show that the two solution function y = e^x and y = xe^x are not constant multiples of each other.

(g) The significance of part (f) is that together with parts (a) and (b) it implies5 that the general solution of equation (⋆) has the form

y = ce^x+ (dxe)^x for any constants c and d.

(h) Use the general solution in part (g) to solve the initial value problem y′′ − 2y′ + y = 0

with initial conditions y(0) = 7 and y′(0) = 4.

Answer 1