What is the general solution of the differential equation dy/dx = 3y?

What is the general solution of the differential equation dy/dx = 3y?

Step 1: Start with the differential equation dy/dx = 3y.
Step 2: Recognize that this equation is separable, meaning we can separate y and x.
Step 3: Rewrite the equation as dy/y = 3 dx.
Step 4: Integrate both sides. The left side becomes ln|y| and the right side becomes 3x + C (where C is the constant of integration).
Step 5: Write the result of the integration: ln|y| = 3x + C.
Step 6: To solve for y, exponentiate both sides to remove the natural logarithm: |y| = e^(3x + C).
Step 7: Rewrite e^(3x + C) as e^(3x) * e^C. Let e^C be a new constant, which we can call C'.
Step 8: Thus, we have |y| = C' * e^(3x). Since C' can be positive or negative, we can write y = C * e^(3x), where C is any constant.

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