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Question
For the differential equation, find the general solution:
y dx + (x – y2) dy = 0
Solution
y dx + (x – y2) dy = 0
or `dx/dy + x/y = y`
This is a linear differential equation of the form `dy/dx + Py = Q.`
Here P = `1/y, Q = y`
∴ `I.F. = e^(int P dx) = e^(int (1/y)dy) = e^(log y) = y`
Hence, the general solution of the differential equation
`x × I.F. = int Q xx (I.F.) dy + C`
⇒ `x xx y = int y xx y dy + C`
⇒ `xy = int y^2 dy + C`
⇒ `xy = 1/3 y^3 + C`
⇒ `x = y^2/3 + C/y`
Which is the required solution.
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