Hints: In general, the solution to x'= Ax is of the form . You need to determine what e^{At} is for matrix A. Find the eigenvalues and eigenvectors (and, if necessary, the generalized eigenvectors) of the coefficient matrix. Use that to diagonalize the matrix or put it into Jordan normal form and then find .

Alternatively, less sophisticated but simpler for a very simple equation, rewrite this as a system of equations: Writing the vector x as (x(t), y(t), z(t)), we have x'= -x+ y, y'= -y, z'= x- z.

From y'= -y, we have and, since y(0)= 1, . Then , a first order linear equation. Once you have solved that, put it into z'= x- z to get a linear first order equation for z.