Solve a Separable Differential Equation with Initial Condition

Solve the differential equation $\frac{dy}{dx} = 3x^2 e^{-y}$ that satisfies the initial condition $y(0) = 0$.

This problem involves solving a type of differential equation known as a separable differential equation. A separable differential equation can be expressed in the form where all terms involving the dependent variable can be written on one side and all terms involving the independent variable on the other side. This allows us to take advantage of direct integration to find a solution. The challenge is typically in correctly separating the variables and integrating appropriately to find the function satisfying the given differential equation, particularly with the initial condition provided.

In this case, the differential equation is $\frac{dy}{dx} = 3x^2 e^{-y}$, and the goal is to separate the variables such that we can integrate both sides. Here, the manipulation involves moving terms to achieve $g(y)dy = h(x)dx$, allowing integration of both sides. After integration, the application of the initial condition $y(0) = 0$ is crucial. Initial conditions are vital as they help in determining any constants of integration that are introduced through the indefinite integrals, ensuring the solution satisfies specific criteria at given points.

It is also important to remember the exponential function's properties when dealing with the integration involving $e^{-y}$. Understanding how to work with exponential and polynomial functions in integration ensures a smooth path toward solving such equations efficiently. This problem highlights the integration techniques and careful handling of algebraic manipulations required to solve initial value problems typical in introductory differential equations courses.

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