Example 27 - Solve (x dy - ydx) y sin (y/x) = (ydx + xdy) - Solving homogeneous differential equation

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  1. Class 12
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Example 27 Solve the differential equation ( ) ( / )=( + ) cos ( / ) ( ) ( / )=( + ) cos ( / ) sin ( / ) ^2 ( / ) = cos ( / ) + ^2 cos ( / ) [ sin ( / ) ^2 ( / ) ] =[ cos ( / )+ ^2 sin ( / ) ] / = ( cos ( / ) + ^2 sin ( / ) )/( sin ( / ) ^2 ( / ) ) Dividing numerator & denominator by x2 / = ( / cos ( / ) + ( / )^2 cos ( / ) )/( / sin ( / ) cos ( / ) ) Let y = vx / = v + / Putting value of / and y in (1) v + ( )/ =( / cos ( / ) + ( ^2 ^2)/ ^2 sin ( / ) )/( / sin ( / ) ( / ) ) v + ( )/ = ( cos + ^2 sin )/( sin cos ) x ( )/ = ( cos + ^2 sin )/( sin cos ) v x ( )/ = ( cos + ^2 sin ( sin cos ) )/( sin cos ) x ( )/ = ( cos + ^2 sin ^2 sin + cos )/( sin cos ) ( )/ = (2 cos )/( sin cos ) (( sin cos )/ v cos ) =2 / (( sin )/ v cos cos / v cos ) dv = 2 / (tan 1/ ) dv = 2 / Integrating both sides 1 (tan 1/ ) =2 1 / 1 tan 1 / = 2 1 / log |sec | | | = 2 log | | + log| | sec ( / ) = C xy

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