2020 TMJC P1 Q8

Show that the differential equation \(\frac{\mathrm{d}y}{\mathrm{d}x}=5\left( x-y \right)\) may be reduced by the substitution \(w=x-y\) to \(\frac{\mathrm{d}w}{\mathrm{d}x}=1-5w\). Hence find the general solution for \(~y\) in terms of \(x\).[6]

Find the particular solution of the differential equation \(\frac{\mathrm{d}y}{\mathrm{d}x}=5\left( x-y \right)\) when \(x=0\) and \(y=-1\).[2]

Sketch the graph of the particular solution found in part (ii), showing clearly the behaviour of the graph as \(x\to \infty \), the coordinates of the axial intercept(s) and equations of any asymptote(s).[3]