Question

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

• A. \((x+y+C)e^{x+y}=0\)
• B. \((x+C)e^{x+y}=0\)
• C. \((x-C)e^{x+y}=0\)
• D. \((x+C)e^{x+y}+1=0\)

Hint:

Substitute u = x + y so the equation becomes separable in u and x.

Answer 1

The Correct Option is D

Step 1: Substitute
Let \(u=x+y\). Then \(\frac{du}{dx}=1+\frac{dy}{dx}\), so the equation becomes \(\frac{du}{dx}=e^{u}\).

Step 2: Separate the variables
\(e^{-u}du=dx\).

Step 3: Integrate
\(-e^{-u}=x+C\), that is \(e^{-(x+y)}=-(x+C)\).

Step 4: Rearrange
Multiply by \(e^{x+y}\): \(1=-(x+C)e^{x+y}\), so \((x+C)e^{x+y}+1=0\).

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