What is the solution to the differential equation dy/dx = -y/x?

What is the solution to the differential equation dy/dx = -y/x?

Step 1: Start with the differential equation dy/dx = -y/x.
Step 2: Recognize that this is a separable equation, meaning we can separate the variables y and x.
Step 3: Rearrange the equation to isolate y on one side and x on the other. This gives us dy/y = -dx/x.
Step 4: Integrate both sides. The left side becomes ∫(1/y) dy and the right side becomes ∫(-1/x) dx.
Step 5: The integral of 1/y is ln|y| and the integral of -1/x is -ln|x|. So we have ln|y| = -ln|x| + C, where C is the constant of integration.
Step 6: Rewrite the equation by exponentiating both sides to eliminate the natural logarithm. This gives us |y| = e^C * |x|^(-1).
Step 7: Let C' = e^C, which is just another constant. So we can write |y| = C'/|x|.
Step 8: Since C' can be positive or negative, we can drop the absolute value and write y = C/x, where C is a constant.

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