1.

Solution of the differential equation x=1+xy^((dy)/(dx) + ((xy)^(2))/(2!) ((dy)/(dx))^(2) + ((xy)^(3))/(3!) ((dy)/(dx))^(3) +……. is

Answer»

`y log_(e)X +C`
`y =(log_(e)x)^(2) + C`
`y=+- SQRT((log_(e)x)^(2) + 2C)`
`xy = x^(y) + k`

Solution :We have `x = e^((xy)(dy)/(dx))`
`RARR logx = xy (dy)/(dx) rArr ydy (log x)/x dx`
On INTEGRATION, we get
`y^(2)2 = (log_(e)x)^(2)/2 + C`
`y^(2) = (log_(e)x)^(2) + C`
Hence, `y = +- sqrt((log_(e)x)^(2) + 2C)`


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