Question

The solution of the differential equation $(1 + y e^{xy}) \, dx + (2y + x e^{xy}) \, dy = 0$ is given by

• A. $x + e^{xy} + y^2 = C$
• B. $y + e^{xy} + x^2 = C$
• C. $e^x + xy + e^y = C$
• D. $e^x - xy + e^y = C$

Hint:

An alternative quick method is grouping terms into exact differentials: $(dx + y e^{xy} dx + x e^{xy} dy) + 2y dy = d(x) + d(e^{xy}) + d(y^2) = 0$. Integrating directly gives $x + e^{xy} + y^2 = C$.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
A first-order differential equation of the form $M(x, y) \, dx + N(x, y) \, dy = 0$ is exact if $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$.
Key Formula or Approach:
The general solution of an exact differential equation is: \[ \int M \, dx \text{ (treating } y \text{ as a constant)} + \int (\text{terms in } N \text{ free from } x) \, dy = C \]

Step 2: Detailed Explanation:
Here, we have: \[ M = 1 + y e^{xy} \] \[ N = 2y + x e^{xy} \]
First, check for exactness: \[ \frac{\partial M}{\partial y} = \frac{\partial}{\partial y}(1 + y e^{xy}) = e^{xy} + xy e^{xy} \] \[ \frac{\partial N}{\partial x} = \frac{\partial}{\partial x}(2y + x e^{xy}) = e^{xy} + xy e^{xy} \]
Since $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$, the differential equation is exact.
Now, integrate $M$ with respect to $x$, keeping $y$ constant: \[ \int (1 + y e^{xy}) \, dx = x + y \left( \frac{e^{xy}}{y} \right) = x + e^{xy} \]
Next, identify terms in $N$ that do not contain $x$: The only term in $N = 2y + x e^{xy}$ free from $x$ is $2y$.
Integrate this term with respect to $y$: \[ \int 2y \, dy = y^2 \]
Combine the two parts to get the general solution: \[ x + e^{xy} + y^2 = C \]

Step 3: Final Answer:
The solution of the differential equation is $x + e^{xy} + y^2 = C$.