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Let a solution y=y(x) of the differential equation xsqrt(x^(2)-1) dy-ysqrt(y^(2)-1)dx=0 satisfy y(2)=(2)/(sqrt3) Statement-1, y(x)=sec(sec^(-1)x-(pi)/(6)) Statement-2 : y(x) is given by (1)/(y)=(2sqrt3)/(x)-sqrt(1-(1)/(x^(2))) |
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Answer» Statement-1 is TRUE, Statement-2 is True, Statement-2 is a correct explanation for Statement-1. `xsqrt(X^(2)-1)DY=ysqrt(y^(2)-1)dx` `rArr""(1)/(ysqrt(y^(2)-1))dy=(1)/(x sqrt(x^(2)-1))dx` `rArr""int(1)/(ysqrt(y^(2)-1))dy=int(1)/(xsqrt(x^(2)-1))dx` `rArr""SEC^(-1)y=sec^(-1)x+C` It is given that `y=(2)/(sqrt3)` when x = 2 `THEREFORE""sec^(-1).(2)/(sqrt3)=sec^(-1)2+CrArr(pi)/(6)=(pi)/(3)+CrArrC=-(pi)/(6)` Putting `C=-(pi)/(6)` in (i), we get `sec^(-1)y=sec^(-1)x-(pi)/(6)"...(ii)"` `rArr""y=sec(sec^(-1)x-(pi)/(6))` so, statement-1 is true. From (ii), we have `cos^(-1)((1)/(y))=cos^(-1)((1)/(x))-(pi)/(6)` `rArr""(1)/(y)=cos{cos^(-1)((1)/(x))-(pi)/(6)}` `rArr""(1)/(y)=cos{cos^(-1)((1)/(x))}cos((pi)/(6))+sin(cos^(-1).(1)/(x))sin(pi)/(6)` `rArr""(1)/(y)=(sqrt3)/(2x)+(1)/(2)sqrt(1-(1)/(x^(2)))` So, statement-2 is false. |
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