Solve the differential equation dy/dx = 6x^2y.

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Question: Solve the differential equation dy/dx = 6x^2y.

Options:

Correct Answer: y = Ce^(2x^3)

Solution:

This is a separable equation. Integrating gives y = Ce^(2x^3).

Solve the differential equation dy/dx = 6x^2y.

Solve the differential equation dy/dx = 6x^2y.

Step 1: Identify the differential equation: dy/dx = 6x^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 = 6x^2 dx.
Step 4: Integrate both sides: ∫(1/y) dy = ∫6x^2 dx.
Step 5: The left side integrates to ln|y|, and the right side integrates to 2x^3 + C (where C is the constant of integration).
Step 6: Write the equation from the integrals: ln|y| = 2x^3 + C.
Step 7: Exponentiate both sides to solve for y: |y| = e^(2x^3 + C).
Step 8: Rewrite e^(C) as a new constant, say C', so |y| = C'e^(2x^3).
Step 9: Since y can be positive or negative, we can drop the absolute value: y = Ce^(2x^3), where C can be any real number.

Separable Differential Equations – This concept involves equations that can be separated into functions of y and functions of x, allowing for integration on both sides.
Integration of Exponential Functions – Understanding how to integrate functions that involve exponentials, particularly in the context of solving differential equations.