Introduction:
The given differential equation is dy/dx = x^2 * x, and we need to find the solution to this equation when y(1) = 3. To solve this differential equation, we will use the method of separation of variables.

Separation of Variables:
The first step in solving a differential equation using separation of variables is to separate the variables, i.e., put all terms involving y on one side of the equation and all terms involving x on the other side. Let's do that:

dy = x^3 dx

Integration:
Once we have separated the variables, we can integrate both sides of the equation. Integrating both sides will eliminate the derivatives and give us an equation with y and x.

∫dy = ∫x^3 dx

Integrating the left side with respect to y gives us y. On the right side, we can use the power rule of integration to integrate x^3.

y = (1/4)x^4 + C

Here, C is the constant of integration.

Applying the Initial Condition:
Next, we will apply the initial condition y(1) = 3 to find the value of the constant C. Substituting x = 1 and y = 3 into the equation, we can solve for C.

3 = (1/4)(1^4) + C
3 = 1/4 + C
C = 3 - 1/4
C = 11/4

Final Solution:
Now that we have the value of the constant C, we can write the final solution to the differential equation.

y = (1/4)x^4 + 11/4

This is the solution to the given differential equation dy/dx = x^2 * x with the initial condition y(1) = 3.