Consider the differential equation 2y'' + ty' − 2y = 14, y(0) = y'(0) = 0. In some instances, the Laplace transform can be used
to solve linear differential equations with variable monomial coefficients. Use Theorem 7.4.1., Theorem 7.4.1 Derivatives of Transforms If F(s) = ℒ{f(t)} and n = 1, 2, 3, . . . , then ℒ{tnf(t)} = (−1)n dn dsn F(s), to reduce the given differential equation to a linear first-order DE in the transformed function Y(s) = ℒ{y(t)}. Solve the first-order DE for Y(s). Then find y(t) = &1{Y(s)}.
