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Question
Solve the differential equation `[e^(-2sqrtx)/sqrtx - y/sqrtx] dx/dy = 1 (x != 0).`
Solution
`[e^(- 2sqrtx)/sqrtx - y/sqrtx] dx/dy = 1`
or `dy/dx = e^(- 2sqrtx)/sqrtx - y/sqrtx` ...(i)
Comparing with `dy/dx + Py = Q`
`P = 1/sqrtx, Q = e^(- 2sqrtx)/sqrtx`
∵ `I.F. = e^(x^(-1/2)) = e^(int 1/sqrtx dx) = e^(2sqrtx)`
Hence, the general solution of the equation,
`y * e^(2sqrtx) = int (e^(- 2sqrtx))/sqrtx * e^(2sqrtx) dx + C`
`y * e^(2sqrtx) = int 1/sqrtx dx + C`
`=> ye^(2sqrtx) = 2sqrtx + C`
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