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For the differential equation y² dx + (x - (1)/(y))dy = 0, if the condition y(1)=1 holds, then the expression for x is:
- 4 - (2)/(y) - (e^(1/y))/(e)
- 3 - (1)/(y) + (e^(1/y))/(e)
- 1 + (1)/(y) - (e^(1/y))/(e)
- 1 - (1)/(y) + (e^(1/y))/(e)
Correct answer: 1 + (1)/(y) - (e^(1/y))/(e)
Solution
The correct option is derived from solving the given differential equation and applying the initial condition y(1)=1, which leads to the expression for x being correctly represented as 1 + rac{1}{y} - rac{e^(1/y)}{e}. This matches the solution obtained through integration and substitution.
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