Solve the differential equation dy/dx = y/x. (2023)

Solve the differential equation dy/dx = y/x. (2023)

Step 1: Identify the equation. We have dy/dx = y/x.
Step 2: Recognize that this is a separable equation. This means we can separate the variables y and x.
Step 3: Rewrite the equation to separate the variables. We can write it as 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: Calculate the integrals. The left side gives us ln|y| and the right side gives us ln|x| + C, where C is the constant of integration.
Step 6: Combine the results. We have ln|y| = ln|x| + C.
Step 7: Exponentiate both sides to eliminate the natural logarithm. This gives us |y| = e^(ln|x| + C) = |x| * e^C.
Step 8: Let e^C be a new constant, which we can call K. So, |y| = K|x|.
Step 9: Remove the absolute value by considering K can be positive or negative. Thus, y = Cx, where C is any constant.

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