91Ó°ÊÓ

A differential equation involves a function and its derivatives, representing how the function changes. Understanding them is essential because many real-world systems are modeled using these equations. In this exercise, the differential equation given is \[ y' = 3x^2 y \].This equation is a first-order differential equation because it involves the first derivative of the function, denoted as \( y' \). The goal is to find a function \( y \) that satisfies this relationship. In our problem, the function \( y = 3 e^{x^3} \), when substituted, holds true to the given differential equation. The idea here is to interpret how change in \( x \) affects \( y \) according to the formula represented by the differential equation.

Chain rule is a fundamental concept in differentiation, especially useful when dealing with composite functions. In simpler terms, if a function is nested inside another, the chain rule helps you differentiate it. For instance, in this exercise, the inner function is \( x^3 \) and the outer function is \( e^u \), where \( u = x^3 \). The derivative of \( e^{x^3} \) with respect to \( x \) is found using the chain rule: first, differentiate \( e^u \) with respect to \( u \), which is \( e^u \), and then multiply by the derivative of \( u \) with respect to \( x \), which is \( 3x^2 \). This gives us \( y' = 9x^2 e^{x^3} \). Understanding the chain rule allows you to take derivatives of more complex functions smoothly.

Verifying a solution to a differential equation involves checking if the proposed function satisfies both the differential equation and any initial conditions. In this problem, verification occurs in two steps:

• First, show that when \( y = 3e^{x^3} \) is differentiated, it satisfies the differential equation given as \( y' = 3x^2 y \). As shown in the solution, this condition is satisfied with \( y' = 9x^2 e^{x^3} \).
• Second, check the initial condition \( y(0) = 3 \). By substituting \( x = 0 \) into \( y \), we get \( y(0) = 3e^{0^3} = 3 \), confirming the initial condition.

By fulfilling both these criteria, \( y = 3e^{x^3} \) is verified as a valid solution to the initial-value problem given the criteria specified in the problem. This process ensures that the solution is not just mathematically consistent but also appropriately initialized.