I've stumbled upon this exercise in my book, and I have no idea how to approach it, because of its variable coefficients. I'd probably have to solve for the homogeneous case, and then find the particular solution, and add them to find the general solution for this ODE.
$$u''(t)-e^tu'(t)-e^tu(t)=1$$
$$u''(t)-e^tu'(t)-e^tu(t)=1$$ $$u''-(e^tu'+e^tu)=1$$ $$u''-(e^tu)'=1$$ Integrate $$u'-e^tu=t+K_1$$ Now it's a first order linear differential equation.
Can you take it from there ?