Question

The solution of the differential equation \(\frac{dy}{dx}+1=e^{x+y}\) is

• A. \(e^{x+y}=x+c\), where \(c\) is integral constant
• B. \((x+c)e^{x+y}-1=0\), where \(c\) is integral constant
• C. \((x+c)e^{x+y}+1=0\), where \(c\) is integral constant
• D. \(e^{x+y}=x+y+c\), where \(c\) is integral constant

Hint:

Put v equal to x plus y so the equation separates, then integrate.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
Let \(v=x+y\), so \(\frac{dv}{dx}=1+\frac{dy}{dx}\).

Step 2: Separate the variables
The equation becomes \(\frac{dv}{dx}=e^{v}\), so \(e^{-v}dv=dx\).

Step 3: Integrate
\(-e^{-v}=x+c\), that is \(1+(x+c)e^{v}=0\). Hence \((x+c)e^{x+y}+1=0\).

Final Answer:
The solution is \((x+c)e^{x+y}+1=0\). \[ \boxed{\text{(C) }(x+c)e^{x+y}+1=0\text{, where }c\text{ is integral constant}} \]