Which of the following is true for y(x) that satisfies the differential equation dy/dx = xy - 1 + x - y; y(0) = 0:
Which of the following is true for y(x) that satisfies the differential equation dy/dx = xy - 1 + x - y; y(0) = 0:
Option 1 -
y(1) = e² - 1
Option 2 -
y(1) = 1
Option 3 -
y(1) = e⁻¹ᐟ² - 1/2
Option 4 -
y(1) = e¹ᐟ² - 1
Asked by Shiksha User
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1 Answer
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Correct Option - 4
Detailed Solution:dy/dx = xy - 1 + x - y = x (y + 1) - (y + 1) = (x - 1) (y + 1)
⇒ ∫ dy / (1 + y) = ∫ (x - 1) dx ⇒ ln (1 + y) = x²/2 - x + C
For y (0) = 0, c = 0. ∴ ln (1 + y) = x²/2 - x
1 + y = e^ (x²/2 - x)
y (1) = e^ (1/2 - 1) - 1 = e^ (-1/2) - 1
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