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?

To understand why the correct answer \( |y + 3| = |x| \) contains absolute values, let's solve the given differential equation and apply the initial condition.

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

This can be rewritten as:
\[ x \frac{dy}{dx} = y + 3 \]
\[ \frac{dy}{dx} = \frac{y + 3}{x} \]

This is a first-order linear differential equation. We can solve it using an integrating factor. The general form of the first-order linear differential equation

Explore how to solve the differential equation x(dy/dx) - y = 3, including the use of integrating factors. Understand why the solution involves absolute values and how initial conditions affect the outcome. Suitable for advanced high school students studying calculus and differential equations.

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