91Ó°ÊÓ

The concept of variable separation is pivotal in solving first-order differential equations. In our given problem, the equation is expressed in the form \( dy - ky \cdot dx = 0 \). The goal here is to separate variables, a crucial step in making the equation integrable. To do so, start by rearranging the equation to isolate all terms involving \( y \) on one side and those involving \( x \) on the other.
To achieve this, divide both sides by \( y \) to obtain:

• \( \frac{dy}{y} = k \, dx \)

The logic behind separating variables is to simplify integration by reducing the complexity of the equation, making it easier to find an analytic solution. Now each side of the equation can be independently integrated with respect to its respective variable.

Once variables are separated, integrating both sides of the equation is the next essential step. Integration is a technique to find a function whose derivative matches the original function. Here, we deal with two integrations:

• Left side: \( \int \frac{dy}{y} \)
• Right side: \( \int k \, dx \)

The integral of \( \frac{1}{y} \) with respect to \( y \) results in the natural logarithm, \( \ln|y| \). Similarly, the integral of the constant \( k \) with respect to \( x \) yields \( kx \).
Do not forget the constant of integration, denoted as \( C_1 \), added to the result of the integral on the right side. Thus, the equation becomes:

• \( \ln|y| = kx + C_1 \)

Understanding these integration techniques empowers you to solve and manipulate equations confidently.

The final step involves resolving the equation to express \( y \) explicitly. The natural logarithm present after integrating can be removed by exponentiating both sides. Raised to the power \( e \), the equation becomes:

• \( e^{\ln|y|} = e^{kx + C_1} \)

Leveraging the property \( e^{\ln|y|} = |y| \), we simplify this to:

• |y| = e^{kx + C_1}

At this point, the exponential function introduces an arbitrary constant \( C \). Represent it as \( C = e^{C_1} \), to simplify our final expression:

• |y| = Ce^{kx}

Dropping the absolute marks, the general solution to the differential equation is:

• \( y(x) = Ce^{kx} \)

The constant \( C \) embodies any particular conditions such as initial values, making the solution adaptable to specific problems. This form shows not only the functionality of exponential solutions but also their elegance in encapsulating the essence of dynamic processes.