Solving a Basic Differential Equation

Solve the differential equation $\frac{dy}{dx} = 5x$.

In this problem, we are working with a differential equation, which is a type of mathematical equation that involves functions and their derivatives. Specifically, we're given a first-order differential equation, where the derivative of a function y with respect to x is proportional to x. Problems like these often appear in the early stages of studying differential equations and serve as a fundamental building block for more complex topics.

The primary concept involved is understanding how to find a function given its rate of change, which is frequently expressed as a differential equation. To solve it, one of the most common techniques is integration. By integrating the right-hand side of the equation with respect to x, we can recover the original function y. This highlights the close relationship between differentiation and integration, often described as inverse processes in calculus.

This problem also serves as an excellent introduction to the method of solving by integration, a crucial approach in handling separable differential equations. You'll want to focus on how integration allows us to 'undo' the differentiation, providing a means to solve equations that model a multitude of real-world phenomena. As you watch the solution, consider the implications of such mathematical techniques in both theoretical and applied contexts.

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