What is the solution to the differential equation dy/dx = (x^2 + 1)y?
Correct Answer: y = Ce^(x^3/3 + x)
- 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: Write the equation as 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: Simplify to y = Ce^(x^3/3 + x), where C = e^C is a new constant.
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