Discontinuous Coefficients. As we will see in Chapter 3, occasions arise when the co...

Discontinuous Coefficients. As we will see in Chapter 3, occasions arise when the coefficient P(x) in a linear equation fails to be continuous because of jump discontinuities. Fortunately, we may still obtain a “reasonable” solution. For example, consider the initial value problem

(a) Find the general solution for

(b) Choose the constant in the solution of part (a) so that the initial condition is satisfied.

(c) Find the general solution for x > 2.

(d) Now choose the constant in the general solution from part (c) so that the solution from part (b) and the solution from part (c) agree at x = 2. By patching the two solutions together, we can obtain a continuous function that satisfies the differential equation except at x = 2, where its derivative is undefined.

(e) Sketch the graph of the solution from x = 0 to x = 5.