91Ó°ÊÓ

A first-order linear differential equation is an equation that involves the first derivative of a function, often written as \(y'\), and the function itself, \(y\). These equations take the general form of \(y' + P(x)y = Q(x)\), where \(P(x)\) and \(Q(x)\) are given functions of \(x\). In the given problem, our equation is \(y' = \frac{1}{3} y\). Notice that this fits the form of a first-order linear differential equation, with \(P(x) = -\frac{1}{3}\) and \(Q(x) = 0\). By understanding this basic structure, we can use specific strategies to solve for \(y\).

Separation of variables is a powerful method used to solve first-order differential equations. The goal is to rearrange the equation so that each variable and its derivative are on different sides of the equation. Our given differential equation \(y' = \frac{1}{3} y\) can be rewritten using separation of variables. First, we express \(y'\) as \(\frac{dy}{dx}\):

\(\frac{dy}{dx} = \frac{1}{3} y\).

Next, we rearrange to isolate \(dy\) and \(dx\) on opposite sides:

\(\frac{1}{y} dy = \frac{1}{3} dx\).

This step is crucial because it simplifies the process of integration, which we will do next.

Integration is the process of finding the integral of a function, which is essentially the opposite of differentiation. To solve the separated equation from the previous step, we need to integrate both sides.

The left side \(\frac{1}{y} dy\) integrates to \(\ln|y|\), and the right side \(\frac{1}{3} dx\) integrates to \(\frac{1}{3} x\):

\(\int \frac{1}{y} dy = \int \frac{1}{3} dx\).

This gives us:

\(\ln|y| = \frac{1}{3} x + C\), where \(C\) is the integration constant.

We then solve for \(y\) by exponentiating both sides to get:

\(y = e^{\frac{1}{3} x + C}\).

The general solution provides all possible solutions to a differential equation, typically with an arbitrary constant that can take any value. For the equation we solved, we obtained:

\(y = e^{\frac{1}{3} x + C}\).

Recognizing that \(e^{C}\) is simply a constant that we can call \(K\), the general solution simplifies to:

\(y = K e^{\frac{1}{3} x}\), where \(K\) is an arbitrary constant.

This represents the family of all solutions to the original differential equation. Each value of \(K\) gives a different specific solution, encompassing every possible scenario for the differential equation \(y' = \frac{1}{3} y\). Integrating this concept solidifies how solutions emerge from initial conditions or specific scenarios within the context of the equation.