Question

$\frac{dy}{dx} = xy + x + y + 1$ has the solution}

• A. $\log(y+1) = x^2 + x + c$
• B. $\log(y+1) = x + c$
• C. $\log(y+1) = -x + c$
• D. $\log(y+1) = \frac{x^2}{2} + x + c$

Hint:

Always look for factorization opportunities in differential equations to see if variables can be separated.

Answer 1

The Correct Option is D

Step 1: Concept
Use the variable separable method after factoring the expression.

Step 2: Meaning
Factor the RHS: $xy + x + y + 1 = x(y+1) + 1(y+1) = (x+1)(y+1)$.

Step 3: Analysis
Separate variables: $\frac{dy}{y+1} = (x+1) dx$. Integrate both sides: $\int \frac{dy}{y+1} = \int (x+1) dx$.

Step 4: Conclusion
$\log(y+1) = \frac{x^2}{2} + x + c$. This matches option (D). Final Answer: (D)