Solve the differential equation dy/dx = 2y + 3. (2023)

Solve the differential equation dy/dx = 2y + 3. (2023)

Step 1: Write the differential equation in standard form: dy/dx - 2y = 3.
Step 2: Identify the integrating factor, which is e^(∫-2dx) = e^(-2x).
Step 3: Multiply the entire equation by the integrating factor: e^(-2x) * dy/dx - 2e^(-2x) * y = 3e^(-2x).
Step 4: The left side of the equation can be rewritten as the derivative of a product: d/dx(e^(-2x) * y) = 3e^(-2x).
Step 5: Integrate both sides with respect to x: ∫d/dx(e^(-2x) * y) dx = ∫3e^(-2x) dx.
Step 6: The left side simplifies to e^(-2x) * y, and the right side integrates to -3/2 * e^(-2x) + C.
Step 7: Solve for y by multiplying both sides by e^(2x): y = Ce^(2x) - 3/2.

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