Question

The general solution of the differential equation x² + y² – 2xy \(\frac {dy}{dx}\) = 0 is (where C is a constant of integration.)

• A. 2(x² – y²) + x = C
• B. x² + y² = Cy
• C. x² – y² = Cx
• D. x² + y² = Cx

Answer 1

The Correct Option is C

Given  x² + y² – 2xy \(\frac {dy}{dx}\) = 0
\(\frac {dy}{dx}\) = \(\frac {x^2 +y^2}{2xy}\)
\(\frac {dy}{dx}\) = \(\frac {x}{2y} + \frac {y}{2x}\)
Let \(\frac {y}{x}\) = v
\(\frac {dy}{dx}\)= v +x \(\frac {dv}{dx}\)
v + x\(\frac {dv}{dx}\) = \(\frac {v}{2}\) + \(\frac 12\)v
x\(\frac {dv}{dx}\)= -v + \(\frac {v}{2}\)+ \(\frac 12\) v
x\(\frac {dv}{dx}\) = -\(\frac {v}{2}\)+ \(\frac 12\)v
∫ \(\frac {2v}{1-v^2}\) dv = ∫ \(\frac {1}x{}\) dx
ln \(\frac {1}{1-v^2}\) = ln Cx
Where C is a arbitrary constant.
Now put v = \(\frac {y}{x}\)
ln \(\frac {1}{1-(\frac {y}{x})^2}\)= ln Cx
\(\frac {x^2}{x^2 - y^2}\) = Cx
x² - y² = Cx is also a constant.
Therefore, the correct answer is (C) x² – y² = Cx