4 Find the solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } - \frac { x ^ { 2 } y } { 1 + x ^ { 3 } } = x ^ { 2 }$$ for which \(y = 1\) when \(x = 0\), expressing your answer in the form \(y = \mathrm { f } ( x )\).
\(5 \quad\) A line \(l _ { 1 }\) has equation \(\frac { x } { 2 } = \frac { y + 4 } { 3 } = \frac { z + 9 } { 5 }\).

1. Find the cartesian equation of the plane which is parallel to \(l _ { 1 }\) and which contains the points \(( 2,1,5 )\) and \(( 0 , - 1,5 )\).
2. Write down the position vector of a point on \(l _ { 1 }\) with parameter \(t\).
3. Hence, or otherwise, find an equation of the line \(l _ { 2 }\) which intersects \(l _ { 1 }\) at right angles and which passes through the point ( \(- 5,3,4\) ). Give your answer in the form \(\frac { x - a } { p } = \frac { y - b } { q } = \frac { z - c } { r }\).
4. Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 y = \sin x$$
5. Find the solution of the differential equation for which \(y = 0\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 } { 3 }\) when \(x = 0\).