Question

The solution of the differential equation \(\frac{dy}{dx} = \frac{x-y}{x+y}\), when \(x = 0\) and \(y = 0\) represents ....

• A. Circle
• B. Ellipse
• C. Hyperbola
• D. Pair of straight Lines

Hint:

This is a homogeneous equation; put y = vx and integrate.

Answer 1

The Correct Option is D

Step 1: Exact Equation Route:
Write $(x+y)dy=(x-y)dx$, i.e. $(x-y)dx-(x+y)dy=0$. Check exactness: $\partial_y(x-y)=-1$ and $\partial_x(-(x+y))=-1$. It is exact.

Step 2: Potential Function:
Integrate $(x-y)$ with respect to $x$: $\dfrac{x^2}2-xy+h(y)$. Differentiate in y: $-x+h'(y)=-(x+y)$, so $h'=-y$, $h=-\dfrac{y^2}2$.

Step 3: Solution:
$\dfrac{x^2}2-xy-\dfrac{y^2}2=C$, or $x^2-2xy-y^2=2C$. Passing through the origin forces $C=0$, giving a homogeneous quadratic, which is a pair of lines. Option (D).

Final Answer:
Option (D). \[ \boxed{\text{(D) Pair of straight lines}} \]