What is the general solution of the differential equation dy/dx = 4y? (2019)

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Question: What is the general solution of the differential equation dy/dx = 4y? (2019)

Options:

Correct Answer: y = Ce^(4x)

Exam Year: 2019

Solution:

The differential equation dy/dx = 4y can be solved using separation of variables, leading to y = Ce^(4x).

What is the general solution of the differential equation dy/dx = 4y? (2019)

What is the general solution of the differential equation dy/dx = 4y? (2019)

Step 1: Start with the differential equation dy/dx = 4y.
Step 2: Separate the variables by dividing both sides by y and multiplying both sides by dx. This gives us (1/y) dy = 4 dx.
Step 3: 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 4: After integrating, we have ln|y| = 4x + C.
Step 5: To solve for y, exponentiate both sides to remove the natural logarithm: |y| = e^(4x + C).
Step 6: Rewrite e^(4x + C) as e^(4x) * e^C. Let e^C be a new constant, which we can call C'. So, |y| = C' * e^(4x).
Step 7: Since C' can be positive or negative, we can drop the absolute value and write y = Ce^(4x), where C is any constant.

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