Find the general solution of the equation dy/dx = 3x^2y.

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Question: Find the general solution of the equation dy/dx = 3x^2y.

Options:

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

Solution:

This is a separable equation. Separating and integrating gives y = Ce^(x^3).

Find the general solution of the equation dy/dx = 3x^2y.

Find the general solution of the equation dy/dx = 3x^2y.

Step 1: Identify the equation dy/dx = 3x^2y. This is a differential equation where y depends on x.
Step 2: Recognize that this is a separable equation, meaning we can separate the variables y and x.
Step 3: Rewrite the equation to separate y and x: dy/y = 3x^2 dx.
Step 4: Integrate both sides. The left side becomes ∫(1/y) dy = ln|y| + C1, and the right side becomes ∫3x^2 dx = x^3 + C2.
Step 5: Combine the constants of integration into a single constant C. So we have ln|y| = x^3 + C.
Step 6: Exponentiate both sides to solve for y. This gives us |y| = e^(x^3 + C).
Step 7: Rewrite e^(x^3 + C) as e^(x^3) * e^C. Let e^C be a new constant, which we can call C. So, |y| = Ce^(x^3).
Step 8: Since y can be positive or negative, we can drop the absolute value, giving us y = Ce^(x^3).

Separable Differential Equations – The equation can be separated into functions of x and y, allowing for integration on both sides.
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