What is the solution to the differential equation dy/dx = xy?

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Question: What is the solution to the differential equation dy/dx = xy?

Options:

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

Solution:

This is separable. Separating and integrating gives y = Ce^(x^2/2).

What is the solution to the differential equation dy/dx = xy?

What is the solution to the differential equation dy/dx = xy?

Step 1: Start with the differential equation dy/dx = xy.
Step 2: Recognize that this equation is separable, meaning we can separate the variables y and x.
Step 3: Rewrite the equation as dy/y = x dx. This separates the variables.
Step 4: Integrate both sides. The left side becomes ∫(1/y) dy and the right side becomes ∫x dx.
Step 5: The integral of 1/y is ln|y|, and the integral of x is (1/2)x^2. So we have ln|y| = (1/2)x^2 + C, where C is the constant of integration.
Step 6: To solve for y, exponentiate both sides to eliminate the natural logarithm: y = e^((1/2)x^2 + C).
Step 7: Rewrite e^C as a new constant, which we can call C'. So we have y = C'e^((1/2)x^2).
Step 8: Finally, we can express C' as C, leading to the final solution: y = Ce^(x^2/2).

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