Consider the linear differential equation The natural first guess for a particu...

Consider the linear differential equation

The natural first guess for a particular solution of this equation is

but this guess does not yield a solution because it is already a solution of the associated homogeneous equation.

Guess a power series and use this guess to find the solution with y(0) = a₀ = 0 and y^, (0) = a₁ = 1. Identify the series you obtain as the Taylor series centered at t = 0 of a familiar function.