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

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

Step 1: Start with the differential equation dy/dx = y.
Step 2: Recognize that this is a separable equation, meaning we can separate the variables y and x.
Step 3: Rewrite the equation as dy/y = dx. This separates the variables.
Step 4: Integrate both sides. The left side becomes ∫(1/y) dy = ln|y|, and the right side becomes ∫1 dx = x + C, where C is the constant of integration.
Step 5: After integrating, we have ln|y| = x + C.
Step 6: To solve for y, exponentiate both sides to remove the natural logarithm: |y| = e^(x + C).
Step 7: Rewrite e^(x + C) as e^x * e^C. Let C' = e^C, which is a positive constant, so |y| = C' * e^x.
Step 8: Since y can be positive or negative, we write y = ±C' * e^x. We can replace ±C' with a new constant C, so we have y = Ce^x.
Step 9: This gives us the general solution of the differential equation.

Separable Differential Equations – This concept involves equations that can be separated into two parts, allowing for integration of both sides.
Integration of Natural Logarithm – Understanding how to integrate and manipulate logarithmic functions is crucial for solving the equation.
Exponential Functions – Recognizing that the solution involves an exponential function is key to arriving at the final answer.