Question

The solution of the differential equation \(\frac{dy}{dx}=-\frac{x}{y}\) is:

• A. \(x^2+y^2=2C\), where C is constant of integration.
• B. \(x-y^2=2C\), where C is constant of integration.
• C. \(x^2+y=2C\), where C is constant of integration.
• D. \(x^2-y^2=2C\), where C is constant of integration.

Answer 1

The Correct Option is A

The given differential equation is \(\frac{dy}{dx}=-\frac{x}{y}\). This is a separable differential equation, which means we can rearrange the terms to separate variables \(x\) and \(y\). Let's proceed with the solution.

First, separate the variables:

\(y\,dy=-x\,dx\)

Integrate both sides:

\(\int y\,dy=\int -x\,dx\)

This yields:

\(\frac{y^2}{2}=-\frac{x^2}{2}+C\), where \(C\) is the constant of integration.

Multiplying the entire equation by 2 to eliminate fractions:

\(y^2=-x^2+2C\)

Rearranging gives:

\(x^2+y^2=2C\)

This matches the correct solution provided: \(\boxed{x^2+y^2=2C}\), where \(C\) is a constant of integration.