N2013 P1 Q10

The variables \(x,y\) and \(z\) are connected by the following differential equations.

\(\frac{\mathrm{d}z}{\mathrm{d}x}=3-2z\)  \(\mathrm{(A)}\)

\(\frac{\mathrm{d}y}{\mathrm{d}x}=z\)           \(\mathrm{(B)}\)

1. Given that \(z<\frac{3}{2}\), solve equation \(\mathrm{(A)}\) to find \(z\) in terms of \(x\).[4]
2. Hence find \(y\) in terms of \(x\).[2]
3. Use the result in part (ii) to show that
   \(\frac{{{\mathrm{d}}^{2}}y}{\mathrm{d}{{x}^{2}}}=a\frac{\mathrm{d}y}{\mathrm{d}x}+b\),
   for constants \(a\) and \(b\) to be determined.[3]
4. * Not in the latest 9758 H2 Math Syllabus
   The result in part (ii) represents a family of curves. Some members of the family are straight lines. Write down the equations of two of these lines. On a single diagram, sketch one of your lines together with a non-linear member of the family of curves that has your line as an asymptote.[4]