Question

The general solution of the differential equation $\frac{dy}{dx}+\frac{x}{y}=0$ is ________.

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

Hint:

Rearrange the equation to group $y$ with $dy$ and $x$ with $dx$.

Answer 1

The Correct Option is B

Step 1: Concept
Variable separable method for differential equations.
Step 2: Analysis
$\frac{dy}{dx} = -\frac{x}{y}$. $y dy = -x dx$.
Step 3: Calculation
Integrate both sides: $\int y dy = -\int x dx$ $\frac{y^2}{2} = -\frac{x^2}{2} + C'$ $\frac{x^2}{2} + \frac{y^2}{2} = C'$ $x^2 + y^2 = 2C' = c$.
Step 4: Conclusion
The solution is $x^2 + y^2 = c$.
Final Answer:(B)