Final answer:
The differential equation dy/dx = 3xy is solved by separating variables, integrating both sides, and then exponentiating to eliminate the natural logarithm, which results in the solution y = Ae^(3/2)x^2, where A is a constant.
Step-by-step explanation:
To solve the differential equation dy/dx = 3xy for y ≠ 0, we can use the method of separating variables. First, we divide both sides by y and multiply by dx to get dy/y = 3x dx. Next, we integrate both sides, ∫ dy/y = ∫ 3x dx, which gives us ln|y| = (3/2)x2 + C, where C is the constant of integration.
To solve for y, we exponentiate both sides to get y = e(3/2)x2 + C. Since eC is also a constant, we can write it as A, giving us y = Ae(3/2)x2.
Therefore, the solution to the differential equation is a. y equals some constant A multiplied by e raised to the power of 3/2 x2.