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To solve this differential equation, we can use the method of integrating factors. The integrating factor is $e^{\int 2x dx} = e^{x^2}$. Multiplying both sides of the differential equation by the integrating factor, we get $e^{x^2} \frac{dy}{dx} = 2xe^{x^2} + e^{x^2}$. Now, we can integrate both sides to get $e^{x^2} y = 2e^{x^2} + C$, where $C$ is the constant of integration. Dividing both sides by $e^{x^2}$, we get $y = x^2 + x + C$, which is the general solution.