Solve the DE separately in each region, without applying the conditions. Then apply the conditions to find the constants of integration.
Each separate solution will involve two unknown constants, for a total of four. The boundary conditions will give two equations for those. Use the fact that the solution function must be differentiable at to find the other two- that is, the two functions and the derivatives of the two functions must be the same at .
Go back and start all over again! Your characteristic equation is which does NOT have roots 1 and -3!
What are the roots of ? What are two independent solutions to ?
Now, with right hand side "1", try a solution of the form y= A, a constant. What "A" satisifies ?
Now, you should have two functions, Y1 and Y2, say, so that Y1 is correct between 0 and (I am assuming you meant " " and not as you had at first) and Y2 is correct for . Y1 will, of course, involve two unknown constants (and don't write them both as "c"!). You can determine those from the initial conditions y(0)= 0 and y'(0)= 0. Once you know those, you know Y1(t) completely and can evaluate it at . Y2 will also involve two unknown constants. You can determine those from and .