Question

Which of the following is an integrating factor of $(x^{2} + y^{2} + 2x)dx + 2ydy = 0$?

• A. $e^{x}$
• B. $x^{2}$
• C. $\frac{1}{x}$
• D. $x$

Hint:

If $(M_y - N_x)/N$ is a function of $x$, the I.F. is $e^{\int f(x) dx}$. If it's a function of $y$, use $(N_x - M_y)/M$.

Answer 1

The Correct Option is A

Step 1: Concept
For an equation $M dx + N dy = 0$, if $\frac{1}{N}(\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x}) = f(x)$, then the integrating factor is $e^{\int f(x) dx}$.

Step 2: Meaning
Identify $M = x^{2} + y^{2} + 2x$ and $N = 2y$. Compute partial derivatives: $\frac{\partial M}{\partial y} = 2y$ and $\frac{\partial N}{\partial x} = 0$.

Step 3: Analysis
Calculate the difference: $\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x} = 2y - 0 = 2y$. Now, divide by $N$: $\frac{1}{2y}(2y) = 1$. This is a function of $x$ alone, $f(x) = 1$.

Step 4: Conclusion
The integrating factor is $e^{\int 1 dx} = e^{x}$. Final Answer: (A)