Let's solve the differential equation step by step: Given: \[ \frac{dy}{dx} = 180y x^{17} \] with \( y \)-intercept \( = 3 \) (i.e., \( y(0) = 3 \)). --- ### **Step 1: Separate Variables** \[ \frac{dy}{dx} = 180y x^{17} \] Divide both sides by \( y \): \[ \frac{1}{y} \frac{dy}{dx} = 180 x^{17} \] Rewrite as: \[ \frac{1}{y} dy = 180 x^{17} dx \] --- ### **Step 2: Integrate Both Sides** Integrate with respect to \( x \): \[ \int \frac{1}{y} dy = \int 180 x^{17} dx \] Left side: \[ \int \frac{1}{y} dy = \ln|y| \] Right side: \[ \int 180 x^{17} dx = 180 \cdot \frac{x^{18}}{18} = 10 x^{18} \] So, \[ \ln|y| = 10 x^{18} + C \] --- ### **Step 3: Solve for \( y \)** Exponentiate both sides to get rid of the natural log: \[ y = e^{10 x^{18} + C} \] \[ y = e^C \cdot e^{10 x^{18}} \] Let \( e^C = A \), a constant: \[ y = A e^{10 x^{18}} \] --- ### **Step 4: Apply the Initial Condition (\( y(0) = 3 \))** \[ y(0) = A e^{10 \cdot 0^{18}} = A e^0 = A = 3 \] So, \[ A = 3 \] --- ### **Final Answer** \[ \boxed{y(x) = 3 e^{10 x^{18}}} \]

The solution involves separating variables and integrating both sides to find the general form of the curve. Applying the initial condition allows us to determine the specific constant, resulting in the explicit equation of the curve.