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

Correct Answer: 3

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–4b = 3