2024 MI P2 Q3

Use the substitution \(w = x + y\), show that the differential equation \(\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{3 - x - y}{1 + x + y}\) can be reduced into \(\frac{\mathrm{d}w}{\mathrm{d}x} = \frac{4}{1 + w}\). Solve this differential equation and deduce that the general solution can be written as \((x + y)^2 = 2(3x - y) + A\), where \(A\) is an arbitrary constant.[5]

Find the particular solution for the above differential equation, given that the curve passes through \(y\)-intercept \((0, 5)\). Hence or otherwise, find the coordinates of the other \(y\)-intercept.[2]