Solving a DE with no initial conditions

I'm having some sort of difficulty on my signals homework. I am given the following problem.

Where u(t) is a unit step function. For whatever reason, most of the problems assigned have no initial conditions, and the examples done in class all had initial conditions. I am aware there are two "types" of ways to solve them,

1. General Solution and Particular Solution (fairly comfortable using this)
2. Zero-Input and Zero-Response.

I did some research and am fairly sure this is not a Zero-Input question. The farthest I can go on this problem is solving it via the General and Particular Solution. I solve that and got the following answer as the Complete solution.

$$y(t)= Ae^{-t} + Be^{-2t} + C$$

I got the exponent values by solving the characteristic equation to for the general solution. I got C as a particular solution since I assumed that the forced response, $$x(t) = 1$$ and that $$x(t)u(t) = 1u(t) = u(t)$$

This could all be wrong. I don't think I am approaching it correctly but since we were only given 1 example to go off of, and the book doesn't mention anything about assumptions with no initial conditions, I am at a loss here.

• We haven't been taught that yet, but we have learned convolution if that helps. – user2079679 Jan 25 '15 at 20:02
• One way is to factor your differential expression as $(D+1)(D+2)y$ (here $D=\frac{d}{dt}$). Then, let $z=(D+2)y$ and first solve $(D+1)z=u$, with integrating factor. When that is done, solve $(D+2)y=z$, using integrating factor again. – mickep Jan 25 '15 at 20:20
• Or use the Laplace transform. I'm guessing your textbook only wants a solution for $t>0$. – Fizz Jan 26 '15 at 6:41