Solve the first-order differential equation dy/dx = y/x.

Solve the first-order differential equation dy/dx = y/x.

Step 1: Identify the equation dy/dx = y/x as a separable differential equation.
Step 2: Rewrite the equation to separate the variables: dy/y = dx/x.
Step 3: Integrate both sides: ∫(1/y) dy = ∫(1/x) dx.
Step 4: The left side integrates to ln|y| and the right side integrates to ln|x| + C, where C is the constant of integration.
Step 5: Write the equation as ln|y| = ln|x| + C.
Step 6: Exponentiate both sides to eliminate the natural logarithm: |y| = e^(ln|x| + C).
Step 7: Simplify the right side: |y| = |x| * e^C. Let C' = e^C, so |y| = C'|x|.
Step 8: Remove the absolute value (considering C' can be positive or negative): y = Cx, where C is a constant.

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