91Ó°ÊÓ

Separation of variables is a method used to solve first-order differential equations. The goal is to rearrange the equation so that all the terms involving one variable are on one side of the equation, and all the terms involving the other variable are on the other side. For the equation provided, which is \( y' = -4y \), we start by separating the variables:

\( \frac{dy}{y} = -4 \, dx \)

Here, we have moved all the terms involving \( y \) to the left side and terms involving \( x \) to the right side. This step is crucial because it sets up the equation for the next phase, integration.

The general solution to a differential equation includes all possible solutions. Once we've separated the variables and integrated both sides, we obtain:

\( \int \frac{1}{y} \ dy = \int -4 \, dx \)

This gives us:

\( \ln |y| = -4x + C \),

where \( C \) is the constant of integration.

To find the general solution for \( y \), we need to isolate it. We do this by exponentiating both sides:

\( y = e^{-4x + C} \).

We then use the property of exponents to rewrite \( C \) as \( e^C \), a new constant which we denote as \( C_1 \):

\( y = C_1 e^{-4x} \).
This form represents the general solution to the differential equation, where \( C_1 \) can be any constant value.

The constant of integration is an arbitrary constant added to the solution of an indefinite integral. This constant accounts for all possible vertical shifts of the function. In the given problem, during the integration step, we obtained:

\( \ln |y| = -4x + C \),

where \( C \) is the constant of integration.

After solving for \( y \), we rewrote \( C \) as a constant exponent \( e^C \), simplifying to \( C_1 \). This constant \( C_1 \) represents any possible initial condition the function might have, spanning an infinite number of solutions.

To summarize, the general solution is:

\( y = C_1 e^{-4x} \)

The constant \( C_1 \) can take any value, making it crucial in determining the specific solution when initial conditions are provided.