2009 ACJC P1 Q9
The variables $x$ and $y$ are connected by the differential equation
$\frac{\text{d}y}{\text{d}x}-1-{{\left( x+y \right)}^{2}}=0$.
(i)
Show that the substitution $u=x+y$ reduces the differential equation to $\frac{\text{d}u}{\text{d}x}-2-{{u}^{2}}=0$.
[2]
(ii)
Hence, solve $\frac{\text{d}y}{\text{d}x}-1-{{\left( x+y \right)}^{2}}=0$, expressing $y$ in terms of $x$, given that $y=2$ when $x=0$.
[4]
(iii)
The variable $A$ is defined to be the product of $x$ and $y$, where both $x$ and $y$are functions of $z$. If $\frac{\text{d}A}{\text{d}z}=3$ when $x=0$, find the value of $\frac{\text{d}x}{\text{d}z}$ when $x=0$.
[3]
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