Solve the differential equation dy/dx = 5 - 2y.

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Question: Solve the differential equation dy/dx = 5 - 2y.

Options:

Correct Answer: y = 5/2 + Ce^(-2x)

Solution:

Rearranging gives dy/(5 - 2y) = dx. Integrating both sides leads to y = 5/2 + Ce^(-2x).

Solve the differential equation dy/dx = 5 - 2y.

Solve the differential equation dy/dx = 5 - 2y.

Step 1: Start with the differential equation dy/dx = 5 - 2y.
Step 2: Rearrange the equation to isolate dy on one side: dy/(5 - 2y) = dx.
Step 3: Now, we need to integrate both sides. The left side is ∫(1/(5 - 2y)) dy and the right side is ∫dx.
Step 4: The integral of the left side, ∫(1/(5 - 2y)) dy, can be solved using substitution. Let u = 5 - 2y, then du = -2 dy, or dy = -du/2.
Step 5: Substitute into the integral: ∫(1/u)(-1/2) du = -1/2 ln|u| + C = -1/2 ln|5 - 2y| + C.
Step 6: The integral of the right side, ∫dx, is simply x + C.
Step 7: Set the two integrals equal to each other: -1/2 ln|5 - 2y| = x + C.
Step 8: Solve for y. First, multiply both sides by -2: ln|5 - 2y| = -2x - 2C.
Step 9: Exponentiate both sides to eliminate the natural log: |5 - 2y| = e^(-2x - 2C).
Step 10: Let K = e^(-2C), so |5 - 2y| = Ke^(-2x).
Step 11: Remove the absolute value by considering two cases: 5 - 2y = Ke^(-2x) or 5 - 2y = -Ke^(-2x).
Step 12: Solve for y in both cases. For the first case, 2y = 5 - Ke^(-2x) gives y = 5/2 - (K/2)e^(-2x).
Step 13: For the second case, 2y = 5 + Ke^(-2x) gives y = 5/2 + (K/2)e^(-2x).
Step 14: The general solution can be written as y = 5/2 + Ce^(-2x), where C is a constant.

Separation of Variables – The method used to solve the differential equation by separating the variables y and x.
Integration – The process of finding the integral of both sides of the equation after separation.
Exponential Functions – Understanding the solution involves an exponential function due to the integration of a linear term.