What is the general solution of the equation y' + 4y = 0?

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Question: What is the general solution of the equation y\' + 4y = 0?

Options:

Correct Answer: y = Ce^(-4x)

Solution:

This is a separable equation, and integrating gives y = Ce^(-4x).

What is the general solution of the equation y' + 4y = 0?

What is the general solution of the equation y' + 4y = 0?

Step 1: Identify the equation. We have y' + 4y = 0.
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. This gives us dy/y = -4 dx.
Step 5: Integrate both sides. The left side becomes ln|y| and the right side becomes -4x + C, where C is the constant of integration.
Step 6: Write the result of the integration. We have ln|y| = -4x + C.
Step 7: Exponentiate both sides to solve for y. This gives us |y| = e^(-4x + C).
Step 8: Simplify the equation. We can write e^C as a new constant, say C', so |y| = C'e^(-4x).
Step 9: Remove the absolute value. This gives us y = Ce^(-4x), where C can be any real number.

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