NOTE: Thus far we have assumed thatp(x)andq(x)iny′+ p(x)y = q(x)are continuous, yet in app...

NOTE: Thus far we have assumed thatp(x)andq(x)iny′+ p(x)y = q(x)are continuous, yet in applications that may not be the case. In particular, the “input”q(x)may be discontinuous. In Example 1, for instance,E(t)inL di/dt + Ri = E(t)may well be discontinuous, such as

We state that in such cases, whereE(t)has one or more jump discontinuities, the solution (11) [more generally, (24) in Section 2.2] is still valid, and can be used in these exercises.

(Mass on an inclined plane)The equationmx ″+ex ′ = mgsinαgoverns the straight-line displacementx(t)of a massmalong a plane that is inclined at an angle′with respect to the horizontal, if it slides under the action of gravity and friction. Ifx(0)= 0 and x′ (0) = 0, solve for x(t). HINT: First, integrate the equation once with respect totto reduce it to a first-order linear equation.