The number of solutions of the equation \(xy\,dx - (x^2 - y^2)\,dy = 0\) with \(y(2) = 3\) is:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → One
The equation is
\( \displaystyle \frac{dy}{dx} = \frac{xy}{x^2 - y^2} \)
Both numerator and denominator have degree 2, so it is a homogeneous differential equation. Put y = vx, so that \(\frac{dy}{dx} = v + x\frac{dv}{dx}\):
\( \displaystyle v + x\frac{dv}{dx} = \frac{v}{1 - v^2} \)
\( \displaystyle x\frac{dv}{dx} = \frac{v}{1 - v^2} - v = \frac{v^3}{1 - v^2} \)
Separating the variables:
\( \displaystyle \int\left(\frac{1}{v^3} - \frac{1}{v}\right)dv = \int\frac{dx}{x} \)
\( \displaystyle -\frac{1}{2v^2} - \log|v| = \log|x| + C \)
Since \(\log|v| + \log|x| = \log|y|\) and \(\frac{1}{v^2} = \frac{x^2}{y^2}\):
\( \displaystyle -\frac{x^2}{2y^2} - \log|y| = C \)
The condition y(2) = 3 means x = 2, y = 3:
\( \displaystyle C = -\frac{4}{18} - \log 3 = -\frac{2}{9} - \log 3 \)
The condition fixes exactly one value of C, so there is exactly one particular solution:
\( \displaystyle \frac{x^2}{2y^2} + \log|y| = \frac{2}{9} + \log 3 \)