1. **Problem statement:** Find the real values of $X$ and $Y$ that satisfy the equation $$X(X+i) + Y(Y - i) + i = 13.$$ 2. **Rewrite the equation:** Expand the terms: $$X^2 + Xi + Y^2 - Yi + i = 13.$$ 3. **Separate real and imaginary parts:** Real part: $$X^2 + Y^2 = 13.$$ Imaginary part: $$X - Y + 1 = 0.$$ 4. **Solve the imaginary part for $Y$:** $$Y = X + 1.$$ 5. **Substitute $Y$ into the real part:** $$X^2 + (X+1)^2 = 13.$$ 6. **Simplify:** $$X^2 + X^2 + 2X + 1 = 13,$$ $$2X^2 + 2X + 1 = 13,$$ $$2X^2 + 2X - 12 = 0,$$ $$X^2 + X - 6 = 0.$$ 7. **Solve quadratic equation:** $$X = \frac{-1 \pm \sqrt{1 + 24}}{2} = \frac{-1 \pm 5}{2}.$$ 8. **Find roots:** $$X_1 = 2, \quad X_2 = -3.$$ 9. **Find corresponding $Y$ values:** For $X=2$, $$Y = 2 + 1 = 3.$$ For $X=-3$, $$Y = -3 + 1 = -2.$$ **Final answers:** $$\boxed{(X,Y) = (2,3) \text{ or } (-3,-2)}.$$