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). 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) and then find y(t) = ℒ−1{Y(s)}. ty'' − y' = 5t2, y(0) = 0 y(t) =