Question

Solve
\[ y e^{x/y} dx = (x e^{x/y} + y^2) dy, \quad y \neq 0 \]

• A. \(e^{x/y} = y - C\)
• B. \(e^{x/y} = y + C\)
• C. \(e^{x/y} = 2y + C\)
• D. \(e^{x/y} = y + 2C\)

Hint:

Recognizing the exact differential form \(d(x/y) = \frac{y\,dx - x\,dy}{y^2}\) is a powerful shortcut that avoids the lengthy substitution \(x = vy\) typical of homogeneous equations.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This differential equation can be simplified by identifying a common differential form.
The presence of terms like \(x/y\) suggests using the quotient rule derivative \(d(x/y) = \frac{y\,dx - x\,dy}{y^2}\).

Step 2: Detailed Explanation:
Let us rearrange the given equation:
\[ y e^{x/y} dx = x e^{x/y} dy + y^2 dy \]
Group the terms involving the exponential function on the left-hand side:
\[ y e^{x/y} dx - x e^{x/y} dy = y^2 dy \]
Factor out \(e^{x/y}\):
\[ e^{x/y} (y dx - x dy) = y^2 dy \]
Divide both sides by \(y^2\) (since \(y \neq 0\)):
\[ e^{x/y} \left( \frac{y dx - x dy}{y^2} \right) = dy \]
Notice that the term in parentheses is the exact differential of the quotient \(\frac{x}{y}\):
\[ d\left(\frac{x}{y}\right) = \frac{y dx - x dy}{y^2} \]
Thus, the equation simplifies to:
\[ e^{x/y} d\left(\frac{x}{y}\right) = dy \]
Let us substitute \(u = \frac{x}{y}\):
\[ e^u du = dy \]
Integrating both sides:
\[ \int e^u du = \int dy \]
\[ e^u = y + C \]
Substitute \(\frac{x}{y}\) back in place of \(u\):
\[ e^{x/y} = y + C \]
This matches Option B.

Step 3: Final Answer:
The solution is \(e^{x/y} = y + C\).