Question

Let the solution curve y = f(x) of the differential equation
\(\frac{dy}{dx} + \frac{xy}{x^2 - 1} = + \frac{ x^4+2x}{\sqrt{1 - x^2}}, \quad x \in (-1, 1)\) pass through the origin. Then
\(\int_{-\frac{\sqrt{3}}{2}}^{\frac{\sqrt{3}}{2}} f(x) \,dx\)is

• A. \(\frac{π}{3}-\frac{1}{4}\)
• B. \(\frac{π}{3} - \frac{\sqrt3}{4}\)
• C. \(\frac{π}{6} - \frac{\sqrt3}{4}\)
• D. \(\frac{π}{6} - \frac{\sqrt3}{2}\)

Answer 1

The Correct Option is B

The correct answer is (B) : \(\frac{π}{3} - \frac{\sqrt3}{4}\)
\(\frac{dy}{dx} + \frac{xy}{x^2 - 1} =  \frac{x^4 +2x}{\sqrt{1 - x^2}}\)
which is first order linear differential equation.
Integrating factor \((I.F.) = e^{∫\frac{x}{x^2−1}dx}\)
\(e^{\frac{1}{2}\ln{|x^2 - 1|}} = \sqrt{|x^2 - 1|}\)
\(=\sqrt{1−x^2} ∵x∈(−1,1)\)
Solution of differential equation 
\(y\sqrt{1−x^2}=∫(x^4+2x)dx=\frac{x^5}{5}+x^2+c\)
Curve is passing through origin, c = 0
\(y=\frac{x^5+5x^2}{5\sqrt{1−x^2}}\)
\(\int_{-\frac{\sqrt{3}}{2}}^{\frac{\sqrt{3}}{2}} \left( \frac{x^5 +5x^2}{5\sqrt{1 - x^2}}\right) \,dx = 0 + 2\int_{0}^{\frac{\sqrt{3}}{2}} \frac{x^2}{\sqrt{1 - x^2}} \,dx\)
\(put\ x= \sin \theta\)
\(dx = \cos \theta\)
\(I = 2\int_{0}^{\frac{\pi}{3}} \frac{\sin^2(\theta) \cdot \cos(\theta)d\theta}{\cos(\theta)} \)
\(\int_{0}^{\frac{\pi}{3}} (1 - \cos2\theta) \,d\theta\)
\(=(\theta −\frac{\sin⁡2 \theta }{2})|_{0}^{\frac{π}{3}}\)
\(= \frac{π}{3}- \frac{\sqrt{3}}{{4}} \)