If \frac{dy}{dx}=2e^{x}y^{3}, y(0)=\frac{1}{2} then what is 4y^{2}(2-e^{x}) equal to ?
- A. 1 ✓
- B. 2
- C. 3
- D. 4
Correct Answer: A. 1
Explanation
Separate variables: \frac{dy}{y^3} = 2e^x dx. Integrating gives -\frac{1}{2y^2} = 2e^x + C. Using y(0)=\frac{1}{2}, we get -\frac{1}{2(1/4)} = 2(1) + C \implies C = -4. Substitute back: -\frac{1}{2y^2} = 2e^x - 4 \implies \frac{1}{4y^2} = 2 - e^x \implies 4y^2(2 - e^x) = 1.
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