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प्रश्न
Solve the differential equation:
`e^(x/y)(1-x/y) + (1 + e^(x/y)) dx/dy = 0` when x = 0, y = 1
उत्तर
`e^(x/y) (1- x/y) + (1 + e^(x/y)) dx/dy = 0`
`Put x = vy
`dx/dy = v + y "dv"/dy`
`:. e^v (1 - v) + (1 + e^v).(v + y "dv"/dy) = 0`
`v(1 + e^v) + y(1 + e^v). (dv)/dy = (v - 1)e^v`
`y(1 + e^v) (dv)/dy = e^v v - e^v - ve^v - v`
`y(1 + e^v) (dv)/(dy) = - (v + e^v)`
`(1 + e^v)/(-(v + e^v)) "dv"/dy = 1/y`
`-int (1 + e^v)/(v + e^v) dv = int 1/y dy`
`log c - log(v + e^v) = log y`
`c/(v + e^v) = y`
`y(v + e^v) = c`
`c = y(v + e^v)`
`c = y(x/y + e^(x"/"y))` ....(1)
when x = 0, y = 1
`c = 1(0 + e^o)`
c = 1
Put c = 1 in equation (1)
`1 = y(x/y + e^(x/y))`
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