Question

The general solution of the differential equation \( \frac{dy}{dx} = xy - 2x - 2y + 4 \) is:

• A. \(\frac{1}{(y-2)^2} = \frac{(x-2)^2}{2} + C\)
• B. \(\log_e|y-2| = \frac{(x-2)^2}{2} + C\)
• C. \((y-2)^2 = \frac{(x-2)^2}{2} + C\)
• D. \(\log_e|y-2| = C\)
• E. \(\log_e|y-2| = (x-2)^2 + C\)

Hint:

In solving separable differential equations, always aim to rearrange the equation to isolate the differentials on opposite sides. Integration then typically leads to a direct relationship or an implicit function defining \(y\) in terms of \(x\).

Answer 1

The Correct Option is B

First, rearrange the differential equation to group terms with \(x\) and \(y\): \[ \frac{dy}{dx} = x(y - 2) - 2(y - 2) \] \[ = (x - 2)(y - 2) \] Separating variables and integrating, we have: \[ \frac{dy}{y - 2} = (x - 2) dx \] Integrate both sides: \[ \int \frac{1}{y-2} dy = \int (x-2) dx \] \[ \log_e|y-2| = \frac{(x-2)^2}{2} + C \] Thus, the integral transforms into a logarithmic relationship between \(y - 2\) and a quadratic expression in \(x - 2\), simplified to match the form of option (B).