The general solution to the given differential equation y' = 3x^2y + x^2 is y = Ce^(x^3 + x), where C is the constant of integration.
To find the general solution of the given first-order linear ordinary differential equation (ODE), y' = 3x^2y + x^2, we can use the method of separation of variables. The ODE can be rewritten as dy/dx = 3x^2y + x^2.
Next, we separate variables by moving all terms involving y to one side and terms involving x to the other. This results in (1/y) dy = (3x^2 + 1) dx.
Now, integrate both sides. The integral of (1/y) with respect to y is ln|y|, and the integral of (3x^2 + 1) with respect to x is x^3 + x + C, where C is the constant of integration.
So, we have ln|y| = x^3 + x + C. To eliminate the absolute value, we can express the solution as y = e^(x^3 + x + C) or y = Ce^(x^3 + x), where C is the constant of integration.