Question

The general solution of \(xdy-ydx=xy\,dy\) is

• A. \(y=Ae^{-xy}\)
• B. \(y=Ae^{x}\)
• C. \(\dfrac{y}{x}=Ae^{x}\)
• D. \(\dfrac{x}{y}+\dfrac{y}{x}=C\)

Hint:

For equations involving differentials such as \[ M(x,y)\,dx+N(x,y)\,dy=0, \] first collect all \(dx\) terms together and all \(dy\) terms together before attempting separation of variables.

Answer 1

The Correct Option is A

Step 1: Rewrite as a homogeneous differential form.
x dy – y dx = xy dy → x dy – xy dy = y dx → x(1–y)dy = y dx → dx/x = (1–y)/y dy = (1/y – 1)dy.

Step 2: Integrate directly.
∫ dx/x = ∫ (1/y – 1)dy → ln|x| = ln|y| – y + C.

Step 3: Rearrange to solve for relationship.
ln|x| – ln|y| = –y + C → ln|x/y| = –y + C → x/y = e^(C)·e^(–y) → x = A y e^(–y) → y e^y = A/x. Alternatively, y = A e^(–xy).

Step 4: Final Answer:
y = A e^(–xy).