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(dy)/(dx)=(4x+y+1)^2...

`(dy)/(dx)=(4x+y+1)^2`

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`(dy)/(dx) = (4x +y +1)^(2)`
put ` 4x +y +1 =u " " therefore 4 + (dy)/(dx) = (du)/(dx)`
` (dy)/(dx) = (du)/(dx) -4 " " therefore (1) " becomes" , (du)/(dx) -4 = u^(2)`
` (du)/(dx) =u^(2)+4 " " therefore 1/(u^(2)+4) du =dx`
Integrating we gets,
` int 1/(u^(2) +2^(2)) du = int dx + c_(1)`
`1/2tan^(-1) (u/2) =x+c_(1)`
` tan^(-1) ((4x +y+1)/(2)) =2x +2c_(1)`
` tan^(-1) ((4x +y+1)/2) =2x +c , " where" c =2c_(1)`
This is the general solution.
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