Question

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

\(\begin{array}{l} \frac{dy}{dx}+\frac{1}{x^2-1}y=\left(\frac{x-1}{x+1}\right)^{1/2},x>1 \end{array}\)

passing through the point (2, √(1/3)). Then √7 y(8) is

• A. \(11+6log_e3\)
• B. 19
• C. \(12-2log_e3\)
• D. \(19-6log_e3\)

Answer 1

The Correct Option is D

\(\begin{array}{l} \frac{dy}{dx}+\frac{1}{x^2-1}y=\sqrt{\frac{x-1}{x+1}},x>1\end{array}\)

Integrating factor \(\begin{array}{l} \text{I.F.}=e^{\int\frac{1}{x^2-1}dx}=e^{\frac{1}{2}\text{In}\left|\frac{x-1}{x+1}\right|} \end{array}\)

Solution of differential equation \(\begin{array}{l} y\sqrt{\frac{x-1}{x+1}}=\int\frac{x-1}{x+1}dx =\int\left(1-\frac{2}{x+1}\right)dx\end{array}\)

\(\begin{array}{l} y\sqrt{\frac{x-1}{x+1}}=x-2\text{In}\left|x+1\right|+C\end{array}\)

Curve passes through (2, √ (1/3) )

\(\begin{array}{l} \frac{1}{\sqrt{3}}\times\frac{1}{\sqrt{3}}=2-2\text{In}3+C \end{array}\)

\(\begin{array}{l} C=2\text{In}3-\frac{5}{3} \end{array}\)

\(\begin{array}{l} y\left(8\right)\times\frac{\sqrt{7}}{3}=8-2\text{In}9+2\text{In}3-\frac{5}{3}\end{array}\)

\(\begin{array}{l} \sqrt{7}\cdot y\left(8\right)=19-6\text{In}3 \end{array}\)