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Solve x dy=(y+x(f(y/x))/(f^(prime)(y/x)...

Solve `x dy=(y+x(f(y/x))/(f^(prime)(y/x)))dx`

Text Solution

Verified by Experts

`xdy = (y+x(f(y//x))/(f^(')(y//x)))`dx
`(dy)/(dx) = y/x+(f(y//x))/(f^(')(y//x))`
Substituting `y//x=v, (dy)/(dx) = v+x(dv)/(dx)`
Given equation transforms to
`v+x(dy)/(dx) =v+(f(v))/(f^(')(v))`
`int(f^(')(v))/(f(v)) = int (dx)/(x)`
`log|f(v)|log|x|+logc`
`|f(v)|=c|x|`
`f(y/x)=cx`
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