Find the general solution of the differential equation dy/dx = 2y.

Find the general solution of the differential equation dy/dx = 2y.

Step 1: Identify the differential equation. We have dy/dx = 2y.
Step 2: Recognize that this is a separable equation, meaning we can separate y and x.
Step 3: Rewrite the equation to separate variables: dy/y = 2 dx.
Step 4: Integrate both sides. The left side becomes ln|y| and the right side becomes 2x + C (where C is the constant of integration).
Step 5: Write the result of the integration: ln|y| = 2x + C.
Step 6: To solve for y, exponentiate both sides to eliminate the natural logarithm: |y| = e^(2x + C).
Step 7: Rewrite e^(2x + C) as e^(2x) * e^C. Let e^C be a new constant, which we can call C (since it is still a constant).
Step 8: Now we have |y| = C * e^(2x). Since C can be positive or negative, we can drop the absolute value: y = C * e^(2x).
Step 9: This is the general solution of the differential equation.

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.
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 derived from the integration process.