Let y = y(x) be the solution of the differential equation dy/dx = (y+1)((y+1)e^(x²/2) - x), 0 < x < 2.1, with y(2) = 0. Then the value of dy/dx at x = 1 is equal to:
Let y = y(x) be the solution of the differential equation dy/dx = (y+1)((y+1)e^(x²/2) - x), 0 < x < 2.1, with y(2) = 0. Then the value of dy/dx at x = 1 is equal to:
Option 1 -
-e²/ (e² + 1)²
Option 2 -
2e² / (1 + e²)²
Option 3 -
e²/ (1 + e²)²
Option 4 -
5e² / (e² + 1)²
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1 Answer
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Correct Option - 3
Detailed Solution:The differential equation is rearranged to dt/dx - xt = -e? ²/², where t = 1/ (y+1).
This is a linear first-order differential equation. The integrating factor (I.F.) is e^ (∫-x dx) = e? ²/².
The solution is t * (I.F.) = ∫ Q (x) * (I.F.) dx + c.
t * e? ²/² = ∫ -e? ²/² * e? ²/² dx + c = ∫ -1 dx = -x + c.
Substituting t = 1/ (y+1) back: e? ²/² / (y+1) = -x + c.
Using the initial condition y (2) = 0:
e? ²/ (0+1) = -2 + c ⇒ c = e? ² + 2.
The solution is e? ²/² / (y+1) = 2 + e? ² - x.
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