What is the solution of the equation dy/dx = 6 - 2y? (2021)

What is the solution of the equation dy/dx = 6 - 2y? (2021)

Step 1: Start with the equation dy/dx = 6 - 2y.
Step 2: Rearrange the equation to isolate dy on one side: dy/(6 - 2y) = dx.
Step 3: Integrate both sides. The left side becomes ∫(1/(6 - 2y)) dy and the right side becomes ∫dx.
Step 4: Solve the integral on the left side. The integral of 1/(6 - 2y) is -1/2 * ln|6 - 2y|.
Step 5: The integral on the right side is simply x + C, where C is the constant of integration.
Step 6: Set the results of the integrals equal to each other: -1/2 * ln|6 - 2y| = x + C.
Step 7: Solve for y. First, multiply both sides by -2 to get ln|6 - 2y| = -2x - 2C.
Step 8: Exponentiate both sides to eliminate the natural logarithm: |6 - 2y| = e^(-2x - 2C).
Step 9: Let K = e^(-2C), so |6 - 2y| = Ke^(-2x).
Step 10: Solve for y: 6 - 2y = ±Ke^(-2x). Choose the positive case for simplicity: 6 - 2y = Ke^(-2x).
Step 11: Rearrange to find y: 2y = 6 - Ke^(-2x), so y = 3 - (K/2)e^(-2x).
Step 12: Replace K/2 with a new constant C to simplify: y = 3 - Ce^(-2x).

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