If x×dy/dx-y= 3 , and x= 1 when y = -2, then the relationship between x and y is ?? a) |y+3| = |x| , b)|x+3| = |y|. the right answer is a, but why does it contain absolute value? Don't use integration

The given differential equation is:

\[
x \frac{dy}{dx} - y = 3
\]

This is a first-order linear differential equation that we can solve directly without using integration to find the relationship between \(x\) and \(y\). Let's start by simplifying and analyzing the equation.

We can rearrange it as follows:

\[
x \frac{dy}{dx} = y + 3
\]

Then,

\[
\frac{dy}{dx} = \frac{y + 3}{x}
\]

To understand why the relationship includes absolute values, we notice that for any solution to the differential equation, the equation must hold for both positive and negative values of \(x\). In this particular case, absolute values appear because the function \(x\) can take both positive and negative values, while maintaining a consistent relationship between \(x\) and \(y\). 

Given the initial condition \( x = 1 \) when \( y = -2 \), you can substitute these values into the final relationship. When solving, you would find that \( |y + 3| = |x| \) ensures the equation is valid for all signs of \(x\) and \(y + 3\). The absolute value reflects the fact that the relationship is symmetric and that flipping the signs of \(x\) or \(y + 3\) doesn't change the validity of the equation.

Therefore, the correct relationship includes absolute values to account for the fact that the magnitude (not just the sign) of \(x\) and \(y + 3\) is what determines the equality.

### Would you like more details, or do you have any questions?

Here are five related questions for further exploration:
1. What is the general solution of a first-order linear differential equation?
2. How does the introduction of initial conditions affect the solution of a differential equation?
3. Why do absolute values often appear in solutions involving square roots?
4. Can you have a solution to a differential equation that does not satisfy a given initial condition?
5. How can phase plane analysis be used to visualize the solutions of differential equations?

**Tip:** When analyzing equations with absolute values, remember that they represent two cases: one where the expression inside is positive and another where it's negative.

Explore the solution to a first-order linear differential equation where x(dy/dx) - y = 3 without using integration. Understand why the relationship |y + 3| = |x| includes absolute values and its implications. Suitable for undergraduate students studying differential equations.

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