JEE MAIN - Mathematics (2022 - 26th July Evening Shift - No. 18)
Suppose $$y=y(x)$$ be the solution curve to the differential equation $$\frac{d y}{d x}-y=2-e^{-x}$$ such that $$\lim\limits_{x \rightarrow \infty} y(x)$$ is finite. If $$a$$ and $$b$$ are respectively the $$x$$ - and $$y$$-intercepts of the tangent to the curve at $$x=0$$, then the value of $$a-4 b$$ is equal to _____________.
Answer
3
Explanation
IF $$ = {e^{-x}}$$
$$y\,.\,{e^{-x}} = - 2{e^{ - x}} + {{{e^{ - 2x}}} \over 2} + C$$
$$ \Rightarrow y = - 2 + {e^{ - x}} + C{e^x}$$
$$\mathop {\lim }\limits_{x \to \infty } \,y(x)$$ is finite so $$C = 0$$
$$y = - 2 + {e^{ - x}}$$
$$ \Rightarrow {\left. {{{dy} \over {dx}} = - {e^{ - x}} \Rightarrow {{dy} \over {dx}}} \right|_{x = 0}} = - 1$$
Equation of tangent
$$y + 1 = - 1(x - 0)$$
or $$y + x = - 1$$
So $$a = - 1,\,b = - 1$$
$$ \Rightarrow a - 4b = 3$$
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