Separable Differential Equation 7A03F2

1. **State the problem:** Solve the differential equation $$\sqrt{1-x^2} \, dy + \sqrt{1-y^2} \, dx = 0$$ using the separable method and find the explicit solution. 2. **Rewrite the equation:** Move terms to isolate differentials: $$\sqrt{1-x^2} \, dy = -\sqrt{1-y^2} \, dx$$ 3. **Separate variables:** Divide both sides by $$\sqrt{1-y^2} \sqrt{1-x^2}$$ to separate variables: $$\frac{dy}{\sqrt{1-y^2}} = -\frac{dx}{\sqrt{1-x^2}}$$ 4. **Integrate both sides:** $$\int \frac{dy}{\sqrt{1-y^2}} = - \int \frac{dx}{\sqrt{1-x^2}}$$ 5. **Recall integral formulas:** $$\int \frac{du}{\sqrt{1-u^2}} = \arcsin u + C$$ 6. **Apply integrals:** $$\arcsin y = - \arcsin x + C$$ 7. **Rewrite the solution:** $$\arcsin y + \arcsin x = C$$ 8. **Express explicitly:** Take sine on both sides: $$\sin(\arcsin y + \arcsin x) = \sin C$$ Use sine addition formula: $$y \sqrt{1-x^2} + x \sqrt{1-y^2} = \sin C$$ 9. **Final explicit form:** $$y \sqrt{1-x^2} + x \sqrt{1-y^2} = K$$ where $$K = \sin C$$ is a constant. This is the implicit solution rearranged explicitly in terms of $$y$$ and $$x$$.