Let $y = y(x)$ be the solution curve of the differential equation$(1 + sinx)\frac{dy}{dx} + (y + 1)c

Question

Let $y = y(x)$ be the solution curve of the differential equation

$(1 + sinx)\frac{dy}{dx} + (y + 1)cosx = 0,y(0) = 0$.

If the curve $y = y(x)$ passes through the point $\left( \alpha,\frac{- 1}{2} \right)$, then a value of $\alpha$ is :

Accepted Answer

Sol. [Key Concept: Linear/separable: $ln(y + 1) = - ln(1 + sin\ x) + C$; use $y(0) = 0$, then the given point.]$$\int\frac{dy}{y + 1} = - \int\frac{\cos x}{1 + \sin x}dx $$$$\Rightarrow \ln(y + 1) = - \ln\left( 1 + \sin x \right) + C $$$$\because y(0) = 0 \Rightarrow C = 0 $$$$\Rightarrow y + 1 = \frac{1}{1 + sinx} $$passes through $\left( \alpha, - \frac{1}{2} \right)$$$\Rightarrow sinx = 1 $$$\Rightarrow x = \frac{\pi}{2}$ (as per options)

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