Question

The general solution of the differential equation $xdy-ydx=0$ is

• A. $x^2-y^2=k$
• B. $xy=k$
• C. $x=ky$
• D. $\log y+\log x=k$

Hint:

Equations of the form \(xdy-ydx=0\) usually become separable after rearranging into \(\frac{dy}{y}=\frac{dx}{x}\).

Answer 1

The Correct Option is B

Step 1: Write the differential equation.
Given equation is
\[ xdy-ydx=0 \] Rearrange:
\[ xdy=ydx \] Step 2: Separate variables.
Divide both sides by \(xy\):
\[ \frac{dy}{y}=\frac{dx}{x} \] Step 3: Integrate both sides.
\[ \int\frac{dy}{y}=\int\frac{dx}{x} \] \[ \log y=\log x + C \] Step 4: Simplify the expression.
\[ \log y-\log x=C \] Using logarithm property
\[ \log\left(\frac{y}{x}\right)=C \] Step 5: Express constant form.
\[ \frac{y}{x}=k \] \[ y=kx \] Multiplying both sides by \(x\):
\[ xy=k \] Step 6: Conclusion.
Thus the general solution of the differential equation is
\[ xy=k \] Final Answer: $\boxed{xy=k}$