1. **State the problem:** Solve the initial value problem $$\frac{dy}{dx} = \frac{y^2 - \cos x \sin x}{x(1 - x^2)}$$ with initial condition $$y(0) = 2$$. 2. **Analyze the differential equation:** This is a first-order ODE. The right side is a function of both $$x$$ and $$y$$. We check if it is separable or can be simplified. 3. **Rewrite the equation:** $$\frac{dy}{dx} = \frac{y^2}{x(1 - x^2)} - \frac{\cos x \sin x}{x(1 - x^2)}$$ 4. **Attempt separation of variables:** Rewrite as $$\frac{dy}{dx} - \frac{y^2}{x(1 - x^2)} = - \frac{\cos x \sin x}{x(1 - x^2)}$$ This is a Bernoulli equation of the form $$\frac{dy}{dx} + P(x) y = Q(x) y^n$$ with $$n=2$$. Here, $$P(x) = 0$$ (since no linear $$y$$ term), but the equation is nonlinear due to $$y^2$$. 5. **Rewrite as Bernoulli:** $$\frac{dy}{dx} = \frac{y^2}{x(1 - x^2)} - \frac{\cos x \sin x}{x(1 - x^2)}$$ 6. **Substitute:** Let $$v = y^{-1}$$, so $$y = \frac{1}{v}$$. Then, $$\frac{dy}{dx} = -\frac{1}{v^2} \frac{dv}{dx}$$. 7. **Substitute into the ODE:** $$-\frac{1}{v^2} \frac{dv}{dx} = \frac{1}{v^2 x(1 - x^2)} - \frac{\cos x \sin x}{x(1 - x^2)}$$ Multiply both sides by $$v^2$$: $$-\frac{dv}{dx} = \frac{1}{x(1 - x^2)} - v^2 \frac{\cos x \sin x}{x(1 - x^2)}$$ 8. **Rearranged:** $$\frac{dv}{dx} + v^2 \frac{\cos x \sin x}{x(1 - x^2)} = - \frac{1}{x(1 - x^2)}$$ 9. **This is a nonlinear ODE in $$v$$.** However, the term $$v^2$$ complicates direct integration. 10. **Check initial condition:** At $$x=0$$, the denominator $$x(1 - x^2) = 0$$, so the equation is singular at $$x=0$$. Therefore, the initial condition $$y(0) = 2$$ is at a singular point, making the problem ill-posed or requiring special treatment. 11. **Conclusion:** The initial value problem cannot be solved by standard methods due to singularity at $$x=0$$. **Final answer:** The initial value problem $$\frac{dy}{dx} = \frac{y^2 - \cos x \sin x}{x(1 - x^2)}$$ with $$y(0) = 2$$ is not solvable by elementary methods because the right side is undefined at $$x=0$$, the initial point. **Slug:** initial value problem **Subject:** differential equations