Introduction 27EXAMPLE 9 A Boundary Value Problem with an Infinite
Number of SolutionsAs in the previous two examples, if the boundary value problem
(18) y" + y = O; y(O) = 0, y(7r) = 0is to have a solution, it must be of the form y = A sin x. Imposing the second
bounda ry condition, y( 7f) = 0, results in the equation A sin 7f = A · 0 = 0,
which is satisfied by all choices of the constant A. So the boundary value
problem (18) has an infinite number of solutions- the one-parameter familyof functions y = Asinx, where A is an arbitrary constant.
In the previous three examples, the differential equation and one boundary
condition were identical. However, the results which we obtained were rad-
ically different because of the second boundary condition. These examples
illustrate the complexities inherent in the study of boundary value problems.
These complexities can, at least partially, be attributed to the interaction
of the boundary conditions with the differential equation. Because of these
inherent complexities, we shall not present any general theory for boundary
value problems.
EXERCISES 1.3
1. Verify that y(x) = c 1 ex
2
is a one-parameter family of solutions of thedifferential equation y' - 2xy = 0 on the interval (-oo, oo ).
- Show that y = c 1 e-x + x^2 - 1 is a one-parameter family of solutions of
the differential equation y' + y = x^2 + 2x - 1 on (-oo, oo ). - Show that y = c 1 e-^2 x + c 2 e^3 x is a two-parameter family of solutions of
the differential equation y" - y' - 6y = 0 on ( -oo, oo).
14. Show that y = (
4
x^2 + c^2 )^2 is a one-parameter family of so lutions of the
differential equation y' = xy^112 on (-oo, oo) and verify that y(x) = 0 is
a singular solution.- Consider the differential equation (19) y" - y = 0.
a. Show that y = c 1 e-x + c 2 ex is a two-parameter family of solutions
of the DE (19).
b. Show that y = k 1 sinh x + k 2 cosh x is also a two-parameter family
of solutions of the DE (19).