Let y=y(x) be the solution of the differential equation (d y)/(~d x)+(2 x)/((1+x²)²) y=x e^(1/((1+x²))); y(0)=0. Then the area enclosed by the curve f(x)=y(x) e^(-1/((1+x²))) and the line y-x=4 is _ _ _ _.

Differential Equations: JEE Main 2024 Mathematics question

JEE Main 2024 · 5 Apr, Evening shift · Q34

JEE Main 20245 Apr, Evening shiftMathematicsMediumDifferential Equations163× since 2002

Let

y=y(x)

be the solution of the differential equation

\frac{\mathrm{d} y}{\mathrm{~d} x}+\frac{2 x}{\left(1+x^2\right)^2} y=x \mathrm{e}^{\frac{1}{\left(1+x^2\right)}} ; y(0)=0.

Then the area enclosed by the curve

f(x)=y(x) \mathrm{e}^{-\frac{1}{\left(1+x^2\right)}}

and the line

y-x=4

is ________.

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