Singular Points. Those values of x for which P (x) in equation (4) is not defined are called singular points of the equation. For example, x = 0 is a singular point of the equation xy +2y = 3x, since when the equation is written in the standard form, y + (2/x)y =3, we see that P(x) = 2/x is not defined at x = 0. On an interval containing a singular point, the questions of the existence and uniqueness of a solution are left unanswered, since Theorem 1 does not apply. To show the possible behavior of solutions near a singular point, consider the following equations.
(a) Show that xy +2y =3x has only one solution defined at x = 0. Then show that the initial value problem for this equation with initial condition y(0) = y0 has a unique solution when y0 = 0 and no solution when y0 ≠ 0.
(b) Show that xy -2y = 3x has an infinite number of solutions defined at x = 0. Then show that the initial value problem for this equation with initial condition y (0) = 0 has an infinite number of solutions.
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