1. **State the problem:** We are given the differential equation $$\frac{dy}{dx} = \frac{10}{x^2 + 1}$$ with the initial condition $$y(0) = 0$$. 2. **Formula and approach:** To find the function $$y(x)$$, we integrate both sides with respect to $$x$$: $$y = \int \frac{10}{x^2 + 1} \, dx + C$$ where $$C$$ is the constant of integration. 3. **Integration:** Recall that $$\int \frac{1}{x^2 + 1} \, dx = \arctan(x) + C$$. 4. **Apply the integral:** $$y = 10 \int \frac{1}{x^2 + 1} \, dx + C = 10 \arctan(x) + C$$. 5. **Use initial condition:** Substitute $$x=0$$ and $$y=0$$: $$0 = 10 \arctan(0) + C = 10 \times 0 + C = C$$ so $$C = 0$$. 6. **Final solution:** $$y = 10 \arctan(x)$$. This function satisfies the differential equation and the initial condition.