Understanding the Role of Absolute Value in Differential Equation Solutions

This article delves into the mathematical reasoning behind the inclusion and exclusion of absolute values in the solutions to differential equations, specifically in the context of the equation y' y. It provides a detailed explanation of why, in certain scenarios, the absolute value can be disregarded.

Solving the Differential Equation y' y

Let's start by solving the differential equation y' y. This is a linear first-order ordinary differential equation that can be solved using the method of separation of variables.

Step 1: Solve the Differential Equation

The given differential equation is:

dy/dt y

Following the method of separation of variables, we separate the variables:

Rewrite the equation: dy/y dt Integrate both sides: ∫(1/y)dy ∫dt This yields: ln|y| t C, where C is the constant of integration. Exponentiate both sides to solve for y: |y| e^(t C), which simplifies to |y| Ke^t, where K e^C is a positive constant. We can express both cases as: y ±Ke^t.

Step 2: Remove the Absolute Value

The equation y ±Ke^t implies that the constant K can be any real number, which includes both positive and negative values. Therefore, we can write:

y Ce^t, where C can be any real number (positive or negative).

Here’s why we can ignore the absolute value:

The constant C can take any real value, which encompasses both positive and negative cases. Hence, y Ke^t covers all possible solutions, as any real number can be represented as C ±K.

Conclusion

In summary, while the solution initially includes an absolute value, we can drop it because the constant C can be any real number, thus covering both the positive and negative cases. The general solution of the differential equation y' y is simply:

y Ce^t, where C ∈ ?.

A More Detailed Explanation

For the equation y' y, if we consider the initial condition y_0, we can separate the variables as follows:

Integrating the left side with the absolute value, we get:

ln|y| x C

Exponentiating both sides, we obtain:

|y| Ke^x, where K e^C

Therefore, the integrals of the equation are:

y Ke^x for y_0 0 y -Ke^x for y_0 0

More simply, one can conclude that the integrals of the equation are given by

y Ke^x

with K being an arbitrary constant.

Why Ignoring the Absolute Value Leads to the Correct General Solution

The above explanation highlights that while the inclusion of the absolute value is mathematically rigorous, ignoring it can lead to the correct general solution. This is because the subsequent mistakes made can inadvertently correct the initial oversight. Specifically:

We can write the solution as y Ce^t where C is any real number. This covers both positive and negative values, as C ±K. The method of solving separable equations, however, might miss the equilibrium solution which is y 0. Equilibrium solutions are missed because we divide both sides of the original separable equation by y, which is undefined when y 0. But when y 0, we have an equilibrium solution. Combining the equilibrium solution with the previously found solution, we get the general solution: y 0, or y tilde;K e^t, with tilde;K ∈ ?

This can be simplified further to write:

y tilde;K e^t, with tilde;K ∈ ?

As seen, the solution obtained is the same if the absolute value was ignored, subsequently corrected for the equilibrium solution.