What is the general solution of the equation y' = 4y + 3?

What is the general solution of the equation y' = 4y + 3?

Step 1: Identify the equation. We have y' = 4y + 3.
Step 2: Rewrite the equation in standard form. This means we want it to look like y' - 4y = 3.
Step 3: Find the integrating factor. The integrating factor is e^(∫-4dx) = e^(-4x).
Step 4: Multiply the entire equation by the integrating factor. This gives us e^(-4x)y' - 4e^(-4x)y = 3e^(-4x).
Step 5: Recognize the left side as the derivative of a product. The left side can be written as d/dx(e^(-4x)y).
Step 6: Integrate both sides. We integrate d/dx(e^(-4x)y) = ∫3e^(-4x)dx.
Step 7: Solve the integral on the right side. The integral of 3e^(-4x) is -3/4 e^(-4x) + C, where C is the constant of integration.
Step 8: Write the equation from the integration. We have e^(-4x)y = -3/4 e^(-4x) + C.
Step 9: Solve for y. Multiply both sides by e^(4x) to isolate y: y = Ce^(4x) - 3/4.
Step 10: Simplify the solution. The final general solution is y = Ce^(4x) + 3/4.

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