Question

The general solution of the differential equation \(\frac{dy}{dx} = \frac{x + y + 1}{x - 3y + 5}\) is

• A. \(3(y - 1)^2 - 2(x + 2)(y - 1) - (x + 2)^2 = c\)
• B. \(x^2 - 3y^2 - 4xy - 2x - 10y = c\)
• C. \(3(y + 1)^2 + 2(x - 2)(y + 1) - (x - 2)^2 = c\)
• D. \(x^2 + 3y^2 + 4xy + 2x + 10y = c\)

Hint:

Try shifting variables to simplify symmetric differential equations; then solve and substitute back.

Answer 1

The Correct Option is A

We use substitution \(x + 2 = X\), \(y - 1 = Y\) so the equation becomes: \[ \frac{dY}{dX} = \frac{X + Y}{X - 3Y} \] Now use variable separable method or homogeneous substitution. Simplifying, integrating both sides yields an implicit relation: \[ 3Y^2 - 2XY - X^2 = c \] Now revert back to original variables to get the final form: \[ 3(y - 1)^2 - 2(x + 2)(y - 1) - (x + 2)^2 = c \]