Find the general solution of the equation y' = 3x^2y.

Find the general solution of the equation y' = 3x^2y.

Step 1: Recognize that the equation y' = 3x^2y is a separable differential equation.
Step 2: Rewrite the equation in a separable form: dy/y = 3x^2 dx.
Step 3: Integrate both sides: ∫(1/y) dy = ∫3x^2 dx.
Step 4: The left side integrates to ln|y|, and the right side integrates to x^3 + C, where C is the constant of integration.
Step 5: Write the equation as ln|y| = x^3 + C.
Step 6: Exponentiate both sides to solve for y: |y| = e^(x^3 + C).
Step 7: Rewrite e^(x^3 + C) as e^(x^3) * e^C. Let C' = e^C, which is a new constant.
Step 8: Thus, y = C'e^(x^3). Since C' can be any real number, we can write it as y = Ce^(x^3), where C is any constant.

Separable Differential Equations – The equation can be separated into functions of x and y, allowing for integration.
Integration Techniques – The solution involves integrating both sides after separation.
Exponential Functions – The solution involves the exponential function, which arises from integrating a linear function of y.