If y(x) is the solution of the differential equation = - 2 (y - 1), then equal.

JEE MainMathematicsModerateDifferential equation

If y(x) is the solution of the differential equation
$\frac{dt}{dx}$ = - 2 (y - 1), then
$\lim_{x \rightarrow \infty}\mspace{2mu} y(x)$ equal.

Show answer and solution

\frac{dy}{dx}

= - 2 x (y - 1)

\frac{dy}{dx}

+ 2 xy = 2x
I.F. =

e^{\int 2xdx} = e^{x^{2}}

Solution

\Rightarrow

y \cdot e^{x^{2}} = \int 2xe^{x^{2}}dx + c

y \cdot e^{x^{2}} = e^{x^{2}} + c

........(1)
Given y(0) = 1

\Rightarrow

1. 1 = 1 + c

\Rightarrow

c = 0

\therefore

y \cdot e^{x^{2}} = e^{x^{2}}

\therefore

\lim_{x \rightarrow \infty}\mspace{2mu} y(x) = \lim_{x \rightarrow \infty}\mspace{2mu}(1) = 1

Source: Grass-G / M-13 Mathematics Paper 4 Nov.docx

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