Question

Suppose y = y(x) be the solution curve to the differential equation
$\frac{dy}{dx}−y=2−e^{−x}$ such that $\lim_{{x \to \infty}}y(x)$
is finite. If a and bare respectively the x – and y – intercepts of the tangent to the curve at x = 0, then the value of a – 4b is equal to _____.

Answer 1

The correct answer is 3
If $= e^{−x}$
$y⋅e^{−x} =−2e^{−x}+\frac{e^{−2x}}{2}+C$
$⇒y=−2+e^{−x}+Ce^x$
$\lim_{{x \to \infty}}$ y(x)is finite so C=0
y = –2 + e – x
$⇒\frac{dy}{dx}=−e^{−x}$
$⇒\frac{dy}{dx}| =−1$
Equation of tangent
y + 1 = –1 (x – 0)
or y + x = –1
So a = –1, b = –1
⇒ a –4 b = 3