Solve the differential equation \( \frac{d y}{d x}=x^{3} y \) \( \mathrm{~A} \quad y=A e^{4 x} \) \( \mathrm{~B} \quad y=A e^{x} \) \( \mathrm{C} \quad y=A e^{\frac{x}{4}} \) \( \mathrm{D} \quad y=A e^{\frac{x^{4}}{4}} \)

To solve the differential equation \( \frac{dy}{dx} = x^3 y \), we can separate the variables. We rewrite it as: \[ \frac{dy}{y} = x^3 dx \] Next, we integrate both sides: \[ \int \frac{dy}{y} = \int x^3 dx \] This gives us: \[ \ln |y| = \frac{x^4}{4} + C \] Exponentiating both sides, we have: \[ |y| = e^C e^{\frac{x^4}{4}} = A e^{\frac{x^4}{4}} \] where \( A = e^C \). Thus, the solution to the differential equation is: \[ y = A e^{\frac{x^4}{4}} \] Looking at the options provided, the correct answer is: \( \mathrm{D} \quad y=A e^{\frac{x^{4}}{4}} \) Now, let’s add some fun facts! The 18th-century mathematician and philosopher, Gottfried Wilhelm Leibniz, co-developed calculus independently from Isaac Newton around the same time, and his notation is still widely used today! Fun fact: Leibniz’s notation for the derivative, \( dy/dx \), is a fundamental part of how we express rates of change! Additionally, this particular type of differential equation is often used in mathematical biology to model populations where the growth rate is proportional to the current size. It’s fascinating how mathematical tools can help us understand and predict real-world phenomena, like how populations grow exponentially in ideal conditions!