Question:
The general solution of \(\frac{dy}{dx} = e^{x+y}\) is:
\(\frac{dy}{dx} = e^{x+y}\) का सामान्य हल क्या है?
Options:
- e^{-x} – e^{-y} = c
- e^{-x} + e^y = c
- e^x + e^{-y} = c
- e^{-x} + e^y = c
- e^{-x} – e^{-y} = c
- e^{-x} + e^y = c
- e^x + e^{-y} = c
- e^{-x} + e^y = c
Correct Answer: c)
Explanation:
Given differential equation:
\[ \frac{dy}{dx} = e^{x+y} = e^x \cdot e^y \]
Separating the variables:
\[ e^{-y} dy = e^x dx \]
Integrating both sides:
\[ \int e^{-y} dy = \int e^x dx \implies -e^{-y} = e^x + c_1 \implies e^x + e^{-y} = c \]
दिया गया अवकल समीकरण:
\[ \frac{dy}{dx} = e^{x+y} = e^x \cdot e^y \]
चरों को अलग करने पर:
\[ e^{-y} dy = e^x dx \]
दोनों पक्षों का समाकलन करने पर:
\[ \int e^{-y} dy = \int e^x dx \implies -e^{-y} = e^x + c_1 \implies e^x + e^{-y} = c \]