To understand why the relationship between $x$ and $y$ contains absolute values without using integration, let’s analyze the given differential equation and the initial condition.

Given:

$x \frac{dy}{dx} - y = 3$ with the initial condition $x = 1$ when $y = -2$.

Rearrange the Equation:

We can rearrange the given differential equation as: $\frac{dy}{dx} = \frac{y + 3}{x}$ This suggests that $y + 3$ and $x$ are proportional to each other.

Analyzing the Proportional Relationship:

The equation $\frac{dy}{dx} = \frac{y + 3}{x}$ implies that any change in $y + 3$ with respect to $x$ is directly proportional to $\frac{1}{x}$.

The direct proportionality implies that the quantity $y + 3$ could scale by a constant multiple of $x$, but whether $y + 3$ increases or decreases with $x$ depends on the sign of the relationship.

Since the solution involves proportionality, and the equation can involve both positive and negative changes, the absolute value ensures that the relationship holds for both increasing and decreasing functions of $x$.

Initial Condition:

Given the initial condition $x = 1$ when $y = -2$: $y + 3 = -2 + 3 = 1$ Since $x = 1$, we see that $|y + 3| = |x| = 1$.

Why Absolute Value:

The absolute value arises because the differential equation could describe a family of curves where $y + 3$ could be either positive or negative as $x$ changes, but the relationship between the magnitudes (absolute values) of $y + 3$ and $x$ remains consistent.

Therefore, the absolute value accounts for the possibility that $y + 3$ might change sign as $x$ changes, ensuring the relationship $|y + 3| = |x|$ holds for all values of $x$.

Conclusion:

The absolute value in the relationship $|y + 3| = |x|$ is essential because it reflects the general solution of the differential equation, which includes both positive and negative branches due to the proportionality and the nature of the initial condition.

Would you like further details or have any questions?

Relative Questions:

1. What does the absolute value of a function represent geometrically?
2. How does the proportional relationship $y + 3$ with $x$ lead to a family of solutions?
3. Can the initial condition $x = 1$, $y = -2$ determine the sign of $y + 3$?
4. What would happen if the equation did not include the absolute value?
5. How does the differential equation suggest symmetry in the solution?

Tip:

Understanding the role of absolute values in differential equations can help when dealing with piecewise functions and ensuring the general solution covers all possible scenarios.