Step 1: Identify the equation. We have y' + 4y = 0, where y' is the derivative of y with respect to x.
Step 2: Rewrite the equation. We can write it as y' = -4y.
Step 3: Recognize that this is a separable equation. We can separate the variables y and x.
Step 4: Rewrite the equation in a separable form: dy/y = -4 dx.
Step 5: Integrate both sides. The left side becomes ∫(1/y) dy = ln|y|, and the right side becomes ∫-4 dx = -4x + C, where C is the constant of integration.
Step 6: Combine the results from the integration. We have ln|y| = -4x + C.
Step 7: Exponentiate both sides to solve for y. This gives us |y| = e^(-4x + C) = e^C * e^(-4x).
Step 8: Let C' = e^C, which is also a constant. So, |y| = C' * e^(-4x).
Step 9: Remove the absolute value. We can write y = ±C' * e^(-4x). We can just call it y = Ce^(-4x), where C can be any constant (positive or negative).
Step 10: Final solution: The solution to the equation y' + 4y = 0 is y = Ce^(-4x).
