Question

The general solution of the differential equation ydx + xdy = 0 is:

• A. xy = C, where C is a constant.
• B. \(\frac{1}{x} + \frac{1}{y} = C\), where C is a constant.
• C. log x, log y = C, where C is a constant.
• D. x + y = C, where C is a constant.

Answer 1

The Correct Option is A

The given differential equation is \(ydx + xdy = 0\). To find the general solution, we start by recognizing this as a first-order homogeneous differential equation. The equation can be rewritten as:

\[\frac{dy}{dx} = -\frac{y}{x}\]

This is a separable differential equation. We can separate the variables by rearranging terms to get:

\[\frac{dy}{y} = -\frac{dx}{x}\]

Integrate both sides:

\[\int \frac{dy}{y} = \int -\frac{dx}{x}\]

Upon integration, we obtain:

\[\ln |y| = -\ln |x| + C_1\]

We can rewrite the right side by factoring as follows:

\[\ln |y| + \ln |x| = C_1\]

This simplifies to:

\[\ln |xy| = C_1\]

Exponentiating both sides to remove the natural logarithm gives:

\[|xy| = e^{C_1}\]

Let \(C = e^{C_1}\), a positive constant, so we have:

\[xy = C\]

This implies the general solution to the differential equation is:

xy = C, where C is a constant.