Suppose y=y(x) solves dydx-y=2-e^-x such that _xy(x) is finite.

Question

Suppose $y=y(x)$ solves $\dfrac{dy}{dx}-y=2-e^{-x}$ such that $\lim_{x	o\infty}y(x)$ is finite. If $a,b$ are the $x$- and $y$-intercepts of the tangent at $x=0$, then $a-4b$ is \_\_\_\_\_\_.

Answer: 3

Accepted Answer

I.F. $=e^{-x}$. $ye^{-x} = -2e^{-x}+\dfrac{e^{-2x}}{2}+C \implies y=-2+e^{-x}+Ce^x$.

Finiteness as $x	o\infty$ requires $C=0$, so $y=-2+e^{-x}$; $y'(0)=-1$.

Tangent at $x=0$ (through $(0,-1)$): $y+1=-x \implies$ $x$-intercept $a=-1$, $y$-intercept $b=-1$.

$a-4b = -1+4 = 3.$

Solutions · 1