Here's step by step solution of the problem.
First, we observe that we have a first order differential equation of the form dy/dx = 18*y*x². This is essentially a separable differential equation and can be solved by separating the variables.
So, we rewrite the equation as dy/y = 18*x² dx. Now our goal is to integrate both sides of the equation.
Integrating, we get that ∫ dy/y = ∫ 18x² dx. Upon solving the integral, we get ln|y| = 6x³ + C, where C is the constant of integration.
As we wish to solve for y, we use properties of logarithms to rewrite ln|y| = 6x³ + C as y = e^(6x³ + C), which simplifies to y = |e^(6x³) * e^C|.
Remember that both e^(6x³) and e^C are constants, so we can absorb |e^C| into a new constant k, rewriting the equation as y = k * e^(6x³).
The problem states that the y-intercept of the curve is 4 which implies that when x=0, y=4. Using these values in the equation y = k * e^(6x³) we find that k=4.
Hence, the solution to the differential equation is y = 4 * e^(6x³), and since y = f(x), we have f(x) = 4 * e^(6x³).