Question

The solution of the differential equation \(y^{2}-x^{2}\frac{dy}{dx}=xy\frac{dy}{dx}\) is ...

• A. \(\frac{y}{x}+\log |y|=c\)
• B. \(\frac{y}{x}+\log |x|=c\)
• C. \(\frac{y^{2}}{x}+\log |y|=c\)
• D. \(\frac{y}{x^{2}}+\log |y|=c\)

Hint:

Write dy/dx as a function of y/x, substitute y equal to vx, and separate the variables.

Answer 1

The Correct Option is A

Step 1: Verify by differentiating
Take $\frac{y}{x}+\log y=c$ and differentiate: $\frac{xy'-y}{x^{2}}+\frac{y'}{y}=0$.

Step 2: Simplify
Multiply by $x^{2}y$: $xyy'-y^{2}+x^{2}y'=0$, so $y'(x^{2}+xy)=y^{2}$.

Step 3: Compare
This is exactly the given equation $y^{2}-x^{2}y'=xyy'$, so the first option is the solution.

Final Answer:
Hence the solution is y/x + log|y| = c. \[ \boxed{\text{(A) }\dfrac{y}{x}+\log |y|=c} \]