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Question

# Let y=y(x) be the solution of the differential equation (1−x2)dydx−xy=1,xϵ(−1,1). if y(o)=0, then y(12) is equal to

A
π63
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B
π3
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C
π6
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D
π3
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Solution

## The correct option is B π6√3(1−x2)dydx−xy=1dydx−x1−x2y=11−x2dydx+P(x)g=Q(x)Solving linear differential equation by calculating I.FI.F=e∫P(x)dx=e∫−x1−x2dx=e12∫d(−x2)1−x2=e1/2ln∣∣1−x2∣∣=eln∣∣1−x2∣∣1/2IF=√1−x2So, y(IF)=∫Q(x)(IF)dxy√1−x2=∫x1−x2√1−x2dxy(√1−x2)=∫x√1−x2dxy(√1−x2)=12sin−1x+Cy(√1−x2)=12sin−1x+Cy(0)=00=12sin−10+CC=0So, y(√1−x2)=12sin−1xAt x=12y=12sin−1(12)√1−(12)2=12×π6×2√3=π6√3

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