Find the general solution of the differential equation dy/dx = 4y.

Find the general solution of the differential equation dy/dx = 4y.

Step 1: Start with the differential equation dy/dx = 4y.
Step 2: Recognize that this is a separable differential equation, meaning we can separate the variables y and x.
Step 3: Rewrite the equation as dy/y = 4 dx. This separates y on one side and x on the other.
Step 4: 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 5: After integrating, we have ln|y| = 4x + C.
Step 6: To solve for y, exponentiate both sides to eliminate the natural logarithm: |y| = e^(4x + C).
Step 7: 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 8: Since C' can be positive or negative, we can drop the absolute value and write y = C * e^(4x), where C is any constant.

Separable Differential Equations – This concept involves equations that can be separated into two parts, one involving y and the other involving x, allowing for integration.
Exponential Functions – Understanding the properties of exponential functions is crucial, as the solution involves the exponential function e raised to a power.
Integration Constants – Recognizing the importance of the constant of integration (C) in the general solution of differential equations.