91Ó°ÊÓ

A differential equation is a mathematical equation that relates some function with its derivatives. In simple terms, it tells us how a function changes over time. Differential equations are an essential tool in modeling real-world systems in science and engineering. They come in various shapes and complexities but are mainly categorized into ordinary differential equations (ODEs) and partial differential equations (PDEs). Our problem involves an ordinary differential equation since it includes only ordinary derivatives of the function.

A first-order linear differential equation is a type of differential equation that can be written in the form \[y' + a(x)y = b(x)\] where

• \(y'\) is the derivative of \(y\) with respect to \(x\)
• \(a(x)\) and \(b(x)\) are functions of \(x\).

In our given exercise, the equation \[y' = 2(10 - y)\] can be rewritten as \[y' + y = 20\] which fits the standard form. Here, \(a(x) = 1\) and \(b(x) = 20\). This kind of equation is straightforward to solve using an integrating factor, which leads us to our next core concept.

The integrating factor is a handy tool for solving first-order linear differential equations. The integrating factor, usually denoted as \(IF\), is a function used to simplify the equation and make it easier to solve. It is given by: \[ IF = e^{\text{integral of } a(x) \text{ with respect to } x} \] For our problem, where \(a(x) = 1\), the integrating factor becomes: \[IF = e^{\text{integral of } 1 \text{ with respect to } x} = e^x \] Multiplying the entire differential equation by \(e^x\) simplifies the left-hand side into the derivative of \(e^x y\), allowing us to integrate both sides easily. This step eventually helps us solve for \(y\) and find the particular solution by applying the initial condition \(y(0) = 1\). Integrating factors make complex differential equations much more manageable.