Differential EquationMedium
Question
Let $y = y(x)$ be the solution of the differential equation $\frac{dy}{dx} = \left( 1 + x + x^{2} \right)\left( 1 - y + y^{2} \right),y(0) = \frac{1}{2}$. Then $(2y(1) - 1)$ is equal to-
Options
A.$\sqrt{3}tan\left( \frac{11\sqrt{3}}{6} \right)$
B.$\frac{\sqrt{3}}{2}tan\left( \frac{11\sqrt{3}}{12} \right)$
C.$\sqrt{3}tan\left( \frac{11\sqrt{3}}{12} \right)$
D.$\frac{\sqrt{3}}{2}tan\left( \frac{11\sqrt{3}}{6} \right)$
Solution
Sol. [Key Concept: Separate: $\int\frac{dy}{y^{2} - y + 1} = \int(x^{2} + x + 1)dx$; apply $y(0) = \frac{1}{2}$.]
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