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Solve `x (dy)/(dx)sin(y/x)+x-ysin(y/x)=0` given `y(1)=pi/2` |
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Answer» `x(dy/dx)sin(y/x)+x-ysin(y/x) = 0` Dividing given equation with `xsin(y/x)` `=>dy/dx+1/sin(y/x)-y/x = 0` Let `y/x = v` Then, `y = vx =>dy/dx = v+x(dv)/dx` So, given equation becomes, `=>v+x(dv)/dx+1/sinv-v = 0` `=> x(dv)/dx = -1/sinv` `=> sinvdv = -dx/x` Integrating both sides, `=> int sinvdv = int -dx/x` `=>-cos v = -ln x +c` `=>-cos(y/x) = -ln x +c` Now, we are given, `y(1) = pi/2` So, putting `x = 1 and y = pi/2` `=>-cos pi/2 = -ln(1) + c => c = 0` So, our equation becomes, `=>-cos(y/x) = -lnx` `=>cos(y/x) = ln x`, which is the required solution. |
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