1. **State the problem:** Solve the equation $$\sqrt{\frac{x}{x+3}} - \sqrt{\frac{x+3}{x}} = 2.$$\n\n2. **Identify domain restrictions:** Since we have square roots and denominators, we require $$x > 0$$ and $$x+3 > 0 \Rightarrow x > -3.$$ Combining, domain is $$x > 0.$$\n\n3. **Rewrite the equation:** Let $$a = \sqrt{\frac{x}{x+3}}$$ and $$b = \sqrt{\frac{x+3}{x}}.$$ The equation becomes $$a - b = 2.$$\n\n4. **Note the product:** $$ab = \sqrt{\frac{x}{x+3} \cdot \frac{x+3}{x}} = \sqrt{1} = 1.$$\n\n5. **Square both sides:** $$(a - b)^2 = 2^2 \Rightarrow a^2 - 2ab + b^2 = 4.$$\n\n6. **Substitute known values:** $$a^2 = \frac{x}{x+3}, \quad b^2 = \frac{x+3}{x}, \quad ab = 1.$$\nSo, $$\frac{x}{x+3} - 2(1) + \frac{x+3}{x} = 4.$$\n\n7. **Simplify:** $$\frac{x}{x+3} + \frac{x+3}{x} - 2 = 4 \Rightarrow \frac{x}{x+3} + \frac{x+3}{x} = 6.$$\n\n8. **Find common denominator and combine:** $$\frac{x^2}{x(x+3)} + \frac{(x+3)^2}{x(x+3)} = 6 \Rightarrow \frac{x^2 + (x+3)^2}{x(x+3)} = 6.$$\n\n9. **Expand numerator:** $$x^2 + (x+3)^2 = x^2 + (x^2 + 6x + 9) = 2x^2 + 6x + 9.$$\n\n10. **Write equation:** $$\frac{2x^2 + 6x + 9}{x(x+3)} = 6.$$\n\n11. **Multiply both sides by denominator:** $$2x^2 + 6x + 9 = 6x(x+3) = 6x^2 + 18x.$$\n\n12. **Bring all terms to one side:** $$2x^2 + 6x + 9 - 6x^2 - 18x = 0 \Rightarrow -4x^2 - 12x + 9 = 0.$$\n\n13. **Multiply entire equation by -1:** $$4x^2 + 12x - 9 = 0.$$\n\n14. **Use quadratic formula:** $$x = \frac{-12 \pm \sqrt{12^2 - 4 \cdot 4 \cdot (-9)}}{2 \cdot 4} = \frac{-12 \pm \sqrt{144 + 144}}{8} = \frac{-12 \pm \sqrt{288}}{8}.$$\n\n15. **Simplify square root:** $$\sqrt{288} = \sqrt{144 \cdot 2} = 12\sqrt{2}.$$\n\n16. **Final solutions:** $$x = \frac{-12 \pm 12\sqrt{2}}{8} = \frac{-12}{8} \pm \frac{12\sqrt{2}}{8} = -\frac{3}{2} \pm \frac{3\sqrt{2}}{2}.$$\n\n17. **Check domain:** Recall $$x > 0.$$\n- For $$x = -\frac{3}{2} + \frac{3\sqrt{2}}{2} = \frac{3}{2}(-1 + \sqrt{2})$$, since $$\sqrt{2} \approx 1.414$$, $$-1 + 1.414 = 0.414 > 0$$, so this solution is valid.\n- For $$x = -\frac{3}{2} - \frac{3\sqrt{2}}{2}$$, this is negative, so discard.\n\n**Final answer:** $$\boxed{x = \frac{3}{2}(-1 + \sqrt{2})}.$$