Find \( f(x) \) if \( y=f(x) \) satisfies \[ \frac{d y}{d x}=18 y x^{8} \] and the \( y \)-intercept of the curve \( y=f(x) \) is 3 \( f(x)= \)

To solve the differential equation \[ \frac{d y}{d x} = 18 y x^{8}, \] we can separate the variables. Rearranging gives us: \[ \frac{dy}{y} = 18x^8 \, dx. \] Now we integrate both sides: \[ \int \frac{1}{y} \, dy = \int 18x^8 \, dx. \] This results in: \[ \ln |y| = 2x^9 + C, \] where \( C \) is the constant of integration. To eliminate the natural logarithm, we exponentiate both sides, giving: \[ |y| = e^{2x^9 + C} = e^C e^{2x^9}. \] Let \( A = e^C \) (where \( A \) is a positive constant), so we have: \[ y = A e^{2x^9}. \] Next, we use the condition that the y-intercept of the curve \( y = f(x) \) is 3. The y-intercept occurs when \( x = 0 \): \[ f(0) = A e^{2 \cdot 0^9} = A. \] Since this y-intercept equals 3, we find that \( A = 3\). Thus, \[ f(x) = 3 e^{2x^9}. \] So the final solution is: \[ f(x) = 3 e^{2x^9}. \]