1. **State the problem:** Solve the differential equation $$y' = 1 + y^2 + \sin x$$ with the initial condition $$y(0) = 0$$. 2. **Understand the equation:** This is a first-order nonlinear ordinary differential equation. The term $$1 + y^2$$ suggests a Riccati-type equation, and the $$\sin x$$ is a nonhomogeneous term. 3. **Rewrite the equation:** $$\frac{dy}{dx} = 1 + y^2 + \sin x$$. 4. **Attempt to solve:** This equation is nonlinear and does not separate easily. However, we can try to find an integrating factor or use substitution methods, but no simple closed-form solution exists in elementary functions. 5. **Check initial condition:** $$y(0) = 0$$. 6. **Numerical or series solution:** Since no elementary closed form is available, the solution can be approximated numerically or expressed as a power series around $$x=0$$. 7. **Summary:** The problem is a nonlinear ODE with initial condition, no simple closed form. Numerical methods or series expansions are recommended for solutions. **Final answer:** The solution to $$y' = 1 + y^2 + \sin x$$ with $$y(0)=0$$ cannot be expressed in elementary functions and requires numerical or approximate methods.