Step 1 :We are given the system of equations: \[\begin{align*} (a+b) x+(a-b) y &= 2 \\ (a^3+b^3) x+(a^3-b^3) y &= 2(a^2+b^2) \end{align*}\]
Step 2 :First, we solve the first equation for x: \[x = \frac{-a*y + b*y + 2}{a + b}\]
Step 3 :Substitute x into the second equation: \[y*(a^3 - b^3) + (a^3 + b^3)*(-a*y + b*y + 2)/(a + b) = 2*a^2 + 2*b^2\]
Step 4 :Solve the resulting equation for y: \[y = \frac{1}{a - b}\]
Step 5 :Substitute y back into the first equation to find the solution for x: \[x*(a + b) + 1 = 2\]
Step 6 :Solve the resulting equation for x: \[x = \frac{1}{a + b}\]
Step 7 :Final Answer: The solutions to the system of equations are \(\boxed{x = \frac{1}{a + b}}\) and \(\boxed{y = \frac{1}{a - b}}\).