Differential-Equations Question 374
Question: The solution of $ \frac{dy}{dx}={2^{y-x}} $ is
[Karnataka CET 2000]
Options:
A) $ 2^{x}+2^{y}=c $
B) $ 2^{x}-2^{y}=c $
C) $ \frac{1}{2^{x}}-\frac{1}{2^{y}}=c $
D) $ x+y=c $
Show Answer
Answer:
Correct Answer: C
Solution:
Given $ \frac{dy}{dx}={2^{y-x}} $ $ =\frac{2^{y}}{2^{x}} $ or $ \frac{dy}{2^{y}}=\frac{dx}{2^{x}} $ Integrating both sides, $ \int{\frac{dy}{2^{y}}=\int{\frac{dx}{2^{x}}}} $ $ -{2^{-y}}\log 2=-{2^{-x}}\log 2+c_1 $ $ \frac{\log 2}{2^{x}}-\frac{\log 2}{2^{y}}=c_1 $ ; $ \frac{1}{2^{x}}-\frac{1}{2^{y}}=\frac{c_1}{\log 2}=c $ .
BETA
