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  • If  is the solution of the differential equation <img alt="\frac{d y}{d x}+\frac{4 x}{\left(x^2-1\right)

If  y=y(x) is the solution of the differential equation \frac{d y}{d x}+\frac{4 x}{\left(x^2-1\right)} y=\frac{x+2}{\left(x^2-1\right)^{\frac{5}{2}}}, x>1such that y(2)=\frac{2}{9} \log _c(2+\sqrt{3}) and  y(\sqrt{2})=\alpha \log _{\mathrm{c}}(\sqrt{\alpha}+\beta)+\beta-\sqrt{\gamma}, \alpha, \beta, \gamma, \in \mathrm{N}, then \alpha \beta \gamma is equal to__________.

Option: 1

6


Option: 2

-


Option: 3

-


Option: 4

-


Answers (1)

best_answer

given differential equation \frac{d y}{d x}+\frac{4 x}{\left(x^2-1\right)} y=\frac{x+2}{\left(x^2-1\right)^{5 / 2}} is linear D.E.
\begin{aligned} & \text { I.F. }=\int_e \frac{4 x}{x^2-1} d x={ }_e 2 \ln \left(x^2-1\right)={ }_e \ln \left(x^2-1\right)^2=\left(x^2-1\right)^2 \\ & y\left(x^2-1\right)^2=\int \frac{x+2}{\left(x^2-1\right)^{5 / 2}}\left(x^2-1\right)^2 d x \\ & =\int \frac{x}{\sqrt{x^2-1}} d x+\int \frac{2 d x}{\sqrt{x^2-1}} \\ & =\sqrt{x^2-1}+2 \ln \left[x+\sqrt{x^2-1}\right]+C \\ & \text { put } y(2)=\frac{2}{9} \ln (2+\sqrt{3}) \end{aligned}

\begin{aligned} & \frac{2}{9} \ln (2+\sqrt{3})(9)=\sqrt{3}+2 \ln [2+\sqrt{3}]+C \\ & =C=-\sqrt{3} \\ & \text { put } x=\sqrt{2} \\ & y=1+2 \ln [\sqrt{2}+1]-\sqrt{3} \\ & \alpha=2, \beta=1=\gamma=3 \\ & \alpha \beta \gamma=2(1)(3)=6 \end{aligned}

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