What is the solution of the differential equation dy/dx = (x^2 + 1)y?

What is the solution of the differential equation dy/dx = (x^2 + 1)y?

Step 1: Identify the differential equation: dy/dx = (x^2 + 1)y.
Step 2: Recognize that this is a separable equation, meaning we can separate y and x.
Step 3: Rewrite the equation as dy/y = (x^2 + 1)dx.
Step 4: Integrate both sides: ∫(1/y) dy = ∫(x^2 + 1) dx.
Step 5: The left side integrates to ln|y|, and the right side integrates to (x^3/3 + x) + C, where C is the constant of integration.
Step 6: Combine the results: ln|y| = (x^3/3 + x) + C.
Step 7: Exponentiate both sides to solve for y: y = e^(x^3/3 + x + C).
Step 8: Rewrite e^C as a new constant, say C, giving the final solution: y = Ce^(x^3/3 + x).

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